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2015

Towards Constraints on the Epoch of : A Phenomenological Approach

Matthew Malloy University of Pennsylvania, [email protected]

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Recommended Citation Malloy, Matthew, "Towards Constraints on the Epoch of Reionization: A Phenomenological Approach" (2015). Publicly Accessible Penn Dissertations. 1875. https://repository.upenn.edu/edissertations/1875

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/1875 For more information, please contact [email protected]. Towards Constraints on the Epoch of Reionization: A Phenomenological Approach

Abstract Based on observations of the early , we know that shortly after the , the Universe was composed almost entirely of neutral and neutral . However, observations of nearby suggest that the between today is neutral to less than one part in 10^4. Thus, it must be the case that some process occurred that stripped the from almost all in the intergalactic medium. Understanding the timing and nature of this process, dubbed ``reionization'', is one of the great outstanding problems in astrophysics and today. In this thesis, we develop several methods for utilizing existing and future measurements in order to make progress toward this end.

We begin by proposing two novel approaches for searching for signatures of underlying neutral hydrogen in the Lya and Lyb forest of distant quasars. We show that, if the Universe is >5% neutral at z ~ 5.5, then damping-wing absorption from neutral hydrogen and absorption from primordial deuterium should leave observable imprints in the Lya and Lyb forest, respectively. Furthermore, the presence of neutral islands should qualitatively alter the size distribution of absorbed regions.

We continue by discussing the ability for the intergalactic medium to retain a thermal memory of the reionization process at z ~ 5, which in turn affects the small-scale structure in the Lya forest. Motivated by this, we model the of the intergalactic medium after reionization and develop a temperature measurement technique that should be able to distinguish between scenarios where reionization ends at z ~ 6 and at z ~ 10.

Lastly, we turn our attention to 21-cm observations during reionization. We demonstrate that, while precise mapping of 21-cm emission from neutral hydrogen should be infeasible by first and second generation interferometers, it may be possible to make crude maps of the reionization process and identify individual ionized regions. This would provide us with direct confirmation that we are observing reionization and provide information regarding its timing and the nature of the ionizing sources.

Degree Type Dissertation

Degree Name Doctor of Philosophy (PhD)

Graduate Group Physics & Astronomy

First Advisor Adam Lidz

Keywords 21-cm, Cosmology, Large Scale Structure, Lyman Alpha Forest, Reionization, Theory

Subject Categories Astrophysics and Astronomy | Physics

This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/1875 TOWARDS CONSTRAINTS ON THE EPOCH OF REIONIZATION:

A PHENOMENOLOGICAL APPROACH Matthew Malloy

A DISSERTATION in Physics and Astronomy Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

2015

Supervisor of Dissertation Graduate Group Chairperson

Adam Lidz Marija Drndi`c Professor, Physics and Astronomy Professor, Physics and Astronomy

Dissertation Comittee:

James Aguirre, Assistant Professor, Physics and Astronomy Cullen Blake, Assistant Professor, Physics and Astronomy Elliot Lipeles, Professor, Physics and Astronomy Masao Sako, Professor, Physics and Astronomy TOWARDS CONSTRAINTS ON THE EPOCH OF REIONIZATION:

A PHENOMENOLOGICAL APPROACH

COPYRIGHT c

2015

Matthew Malloy

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/ To JM

iii Acknowledgements

This thesis truly would not have been possible without a great number of people. First and foremost, I would like to thank my advisor, Adam Lidz, whose endless ability to find interesting and important problems made this an exciting and rewarding experience and whose approach toward tackling those problems has invaluably and irreversibly affected my own. I could not have asked for a better advisor. I would also like to thank collaborators Judd Bowman and Piyanat “Boom” Kittiwisit at ASU, who gave me a greater appreciation for things on the experimental side, and also Andrei Mesinger, Ian McGreer, and Valentina D’Odorico. During my time at Penn, I benefited immensely from conversations and time spent with Garrett Goon and Rami Vanguri. Additionally, I would like to thank the Machine Learning Club and Alan Meert, who taught me literally half of what I have learned in the last year. Ross Anderson, Devin Kennedy, and Miles Wheeler – your inexplicable eagerness to contribute with algorithmic consults was both helpful and touching. I would also like to thank Jessie Taylor for valuable and interesting discussions. I am very fortunate to have been in Philadelphia at the same time as Caitlin Beecham, Chris Bryan, Tom Caldwell, Susan Fowler, Donnie Galvano, Bennet Huber, Andrew Mc- Carthy, Tom Pacific, Doug Schaefer, Katie Schmaling Meert, Elizabeth Stokes, Biquan Su, Debra Van Camp, and Haotian Xian, who, among many other things, played an essential role in me maintaining my sanity. I would also like to thank my parents and my sister who have been a constant source of love and support from the very beginning. Last, but not least, I would like to thank Dingding Jia for her unwavering support and encouragement.

iv ABSTRACT

TOWARDS CONSTRAINTS ON THE EPOCH OF REIONIZATION: A PHENOMENOLOGICAL APPROACH Matthew Malloy Adam Lidz

Based on observations of the early Universe, we know that shortly after the Big Bang, the Universe was composed almost entirely of neutral hydrogen and neutral helium. However, observations of nearby quasars suggest that the gas between galaxies today is neutral to less than one part in 104. Thus, it must be the case that some process occurred that stripped the electrons from almost all atoms in the intergalactic medium. Understanding the timing and nature of this process, dubbed “reionization”, is one of the great outstanding problems in astrophysics and cosmology today. In this thesis, we develop several methods for utilizing existing and future measurements in order to make progress toward this end. We begin by proposing two novel approaches for searching for signatures of underlying neutral hydrogen in the Ly α and Ly β forest of distant quasars. We show that, if the Universe is & 5% neutral at z 5.5, then damping-wing absorption from neutral hydrogen ∼ and absorption from primordial deuterium should leave observable imprints in the Ly α and Ly β forest, respectively. Furthermore, the presence of neutral islands should qualitatively alter the size distribution of absorbed regions. We continue by discussing the ability for the intergalactic medium to retain a thermal memory of the reionization process at redshifts z 5, which in turn affects the small- ∼ scale structure in the Ly α forest. Motivated by this, we model the temperature of the intergalactic medium after reionization and develop a temperature measurement technique that should be able to distinguish between scenarios where reionization ends at z 6 and ∼ at z 10. ∼

v Lastly, we turn our attention to 21-cm observations during reionization. We demonstrate that, while precise mapping of 21-cm emission from neutral hydrogen should be infeasible by first and second generation interferometers, it may be possible to make crude maps of the reionization process and identify individual ionized regions. This would provide us with direct confirmation that we are observing reionization and provide information regarding its timing and the nature of the ionizing sources.

vi Contents

Title i

Copyright ii

Dedication iii

Acknowledgements iv

Abstract v

Contents vii

List of Tables xi

List of Figures xii

1 First Things First 1 1.1 CosmicContext...... 1 1.2 TheShouldersofGiants ...... 7 1.2.1 The Ly α Forest ...... 7

1.2.1.1 Evolution of τeff ...... 14 1.2.1.2 Dark Pixel Covering Fraction ...... 20 1.2.1.3 Damping Wing Redward of Ly α ...... 24

vii CONTENTS

1.2.1.4 IGMTemperature ...... 30 1.2.2 The21-cmLine...... 37 1.2.2.1 The Intensity of the 21-cm Line ...... 38 1.2.2.2 21-cm Fluctuations with Interferometers ...... 44 1.2.2.3 Brief Description of 21-cm Interferometric Experiments . . 54 1.2.2.4 The Global 21-cm Signal ...... 60 1.2.3 The Cosmic Microwave Background ...... 65

1.2.3.1 Optical Depth, τe ...... 66 1.2.3.2 Kinetic Sunyaev-Zel’dovich Effect ...... 70 1.2.4 Ly α Emitters...... 72 1.2.4.1 Clustering of Ly α Emitters...... 72 1.2.4.2 Ly α EmitterFraction ...... 75 1.2.5 Function Measurements ...... 76 1.3 MovingForward ...... 80

2 How to Search for Islands of Neutral Hydrogen in the z 5.5 IGM 82 ∼ 2.1 Introduction...... 82 2.2 Viability of Transmission Through a Partially Neutral IGM ...... 86 2.3 SimulationsandMockSpectra ...... 89 2.4 DarkGapStatistics ...... 94 2.5 StackingToySpectra...... 97 2.5.1 HIDampingWing ...... 98 2.5.2 Deuterium...... 101 2.6 StepsofApproach ...... 104 2.7 Results...... 107 2.7.1 Detecting the Damping Wing ...... 108 2.7.2 DeuteriumFeatureResults ...... 111 2.7.3 DarkGapStatistics ...... 115

viii CONTENTS

2.8 Forecasts ...... 116 2.8.1 Deuterium...... 117 2.8.2 HIDampingWing ...... 118 2.9 Conclusion ...... 119

3 Preliminary Stacking Results 135

4 On Modelling and Measuring the Temperature of the z 5 IGM 143 ∼ 4.1 Introduction...... 143 4.2 Simulations ...... 146 4.3 Reionization Histories ...... 147 4.4 TheThermalStateoftheIGM ...... 153 4.4.1 ModelingtheThermalState ...... 155 4.4.2 Simulated Temperature Field ...... 158 4.4.3 Variations around Fiducial Parameters ...... 167 4.5 Measuring the Temperature of the z 5IGM...... 169 ∼ 4.5.1 Hydrodynamic Simulations: Perfect Temperate-Density Relation Mod- els ...... 172 4.5.2 Degeneracy with the Mean Transmitted Flux ...... 174 4.5.3 Wavelet Amplitude PDFs in Inhomogeneous Reionization Models . . 177 4.5.4 Forecasts ...... 181 4.6 Conclusions ...... 185

5 Identifying Ionized Regions in Noisy Redshifted 21-cm Observations 199 5.1 Introduction...... 199 5.2 Method ...... 202 5.2.1 The21cmSignal...... 202 5.2.2 Semi-Numeric Simulations ...... 203 5.2.3 Redshifted 21 cm Surveys and Thermal Noise ...... 204

ix CONTENTS

5.2.4 Foregrounds...... 207 5.3 ProspectsforImaging ...... 208 5.3.1 TheWienerFilter ...... 208 5.3.2 Application to a Simulated 21 cm Signal ...... 212 5.4 Prospects for Identifying Ionized Regions ...... 213 5.4.1 The Optimal Matched Filter ...... 214 5.4.2 Application to Isolated Spherical Ionized Regions with Noise . . . . 215 5.4.3 Application to a Simulated 21 cm Signal ...... 218 5.4.4 Success of Detecting Ionized Regions ...... 222 5.4.5 Range of Template Radius Considered ...... 224 5.5 Variations on the Fiducial Model ...... 226 5.5.1 IonizedFraction ...... 226 5.5.2 TimingofReionization...... 228 5.5.3 Effects of Foreground Cleaning ...... 229 5.5.4 128 Antenna Tile Configurations ...... 230 5.6 Favorable Antenna Configurations for Bubble Detection ...... 231 5.7 ComparisonstoPreviousWork ...... 233 5.8 Conclusion ...... 234

6 Conclusion 248

Glossary 252

References 256

x List of Tables

3.1 Overview of spectra used in our preliminary stacking tests...... 136

xi List of Figures

1.1 Milestones in the evolution of the Universe from the Big Bang to today.(Photo fromNASAWMAPscienceteam) ...... 3 1.2 Slices through numerical simulations of reionization. The above panels are simulation outputs from McQuinn et al. (109) showing four different reion- ization models. Neutral regions are shown in black and ionized regions are shown in white. Each row is at fixed x with x = 0.8 (top), 0.5 (mid- h HIi h HIi dle), 0.3 (bottom). The of the ionizing sources are related to their by N˙ m1/3 (left), N˙ m (left-middle), N˙ m5/3 (right- ∝ ∝ ∝ middle), and N˙ m but with a larger minimum mass (right). Each slice has ∝ a sidelength of L =93Mpc...... 6 1.3 Illustration of the basic physics behind the Ly α forest and how gas at dif- ferent locations along the line of sight results in absorption lines at different . (Image from http://www.astro.ucla.edu/)...... 12 1.4 Flux as a function of rest-frame for a quasar at z = 0.158 (top) and z = 3.62 (bottom). The denser IGM at higher z results in a dense “forest” of absorption lines blueward of the rest-frame Ly α line (1216A)˚ in the lower panel. (Image from http://www.astro.ucla.edu/)...... 13

xii LIST OF FIGURES

1.5 The inferred evolution of the photoionization rate, ΓHI (left), and neutral fraction (right) from Fan et al. (50). In the left-hand panel, measurements of the effective optical depth in the Ly α (blue), Ly β (green), and Lyγ (ma- genta) forest are converted to estimates of the photoionization rate, with arrows indicating upper bounds. The small circles are measurements in in- dividual bins over the 19 quasars used with the large circles being averages. In the right-hand panel, measurements of the photoionization rate are converted to estimates of the volume-averaged neutral fraction. . . . . 18 1.6 Current limits on x derived from the dark-pixel covering fraction in Mc- h HIi Greer et al. (104). Lightly-shaded points are older limits obtained in McGreer etal.(105)...... 23 1.7 Quasar ULAS J1120+0641 identified at redshift z = 7.085 along with several fitsforthedampingwing...... 27 1.8 of GRB140515A, a -ray burst located at z = 6.33. The right-hand panel overlays damping wing models from a host absorber (blue), a pure IGM model with x = 0.056 (red), and a combination model h HIi (green). The authors argue that, while each curve provides an equally-good fit to the data, the sharp rise in transmission shown is inconsistent with a significantly-neutralIGM...... 29 1.9 Schematic representation of Doppler broadening. The HI is moving away with velocity v from incoming radiation with frequency ν. The observed frequency of the radiation in the atom’s rest frame is ν(1 ξ/c) where ξ is − the component of the velocity parallel with the incident radiation...... 35

xiii LIST OF FIGURES

1.10 Measuring the temperature of the IGM in z & 6 quasar proximity zones. This figure shows mock spectra, and corresponding simulated IGM properties, from Bolton et al. (18) in the top four panels. The bottom panel shows the observed spectrum from SDSS J0818+1722, which Bolton et al. (18) use in order to make temperature measurements inside the proximity zone. Dashed lines indicate regions where Voigt-profile fitting was performed and downward arrows indicate the detected centers of the Voigt profiles...... 36 1.11 Schematic representation of the 21-cm transition where the transition be- tween aligned spins of the and to anti-aligned spins results in the emission of a with λ =21cm...... 42 1.12 Simulation cube of the 21-cm signal during reionization (top-left) along with simulated noise for an interferometer (top-right) and the galactic foregrounds (bottom). This figure demonstrates that, while the sources of noise are sev- eral orders of magnitude larger than the signal, these three contributions to observations are dominant on different scales. The volume of each cube is 1 (Gpc/h)3. In this figure, the line of sight direction away from the observer is to the right and slightly out of the page...... 43 1.13 Depiction of the extra path length, ∆ℓ, of radiation (dot-dashed lines) in- cident on two elements (solid black rectangles) in an array separated by ~r when considering a position on the sky θˆ...... 51

xiv LIST OF FIGURES

1.14 Percentage of pixels “imaged” (SNR > 1) as a function of wavemode, k for the MWA (dashed), LOFAR (dot-dashed), and the SKA (solid). The vertical hatched line shows the distance scale above which (smaller k) the residuals from foreground subtraction are expected to dominate the 21-cm signal. This demonstrates that, for first-generation 21-cm experiments, a very small frac-

tion of pixels with k > khatched will be “imaged”. This estimate assumes that fluctuations in the 21-cm signal are driven from density fluctuations rather than fluctuations in the field, so it is somewhat conservative. Taken fromMcQuinnetal.(113)...... 52 1.15 The redshift evolution of the 21-cm power spectrum in simulated models of reionization. The left panel shows the evolution of the power spectrum during reionization for the fiducial reionization model in Lidz et al. (90). We can see that, as reionization progresses, the slope of the power spectrum in the k-mode range accessible to interferometers (0.1 h/Mpc k 1 h/Mpc) ≤ ≤ declines. The amplitude of this part of the power spectrum peaks around x = 0.5. The right-hand panel shows the evolution of the power spectrum h HIi slope (top) and magnitude (bottom) during reionization for a few different reionization models. This demonstrates that the general power-spectrum evolution described is generic to many reionization models. Both figures are takenfromLidzetal.(90)...... 53 1.16 Several antennae in the GMRT core. www.mso.anu.edu.au ...... 54 1.17 A highly-redundant configuration of tiles for the PAPER interferometer, well- suited for power-spectrum measurements. Picture from www.discovermagazine.com ...... 55 1.18 Several antenna elements in the core of the MWA array. Image taken from www.mwatelescope.org/multimedia...... 56 1.19 Planned layout of the HERA interferometer. Image taken from (45). . . . . 58

xv LIST OF FIGURES

1.20 The central antenna stations for the LOFAR interferometer. Image taken from www..nl...... 59 1.21 An artists impression of what the reionization-focused element of the SKA might look like. “SKA sparse array big” by SKA Project Development Office and Swinburne Astronomy Productions - Swinburne Astronomy Productions for SKA Project Development Office. Licensed under CC BY-SA 3.0 via WikimediaCommons...... 59 1.22 Schematic representation of the 21-cm signal. The top panel shows a plausible signal for 21-cm fluctuations from shortly after the big bang (left) to today (right). Blue indicates the signal is seen in absorption and red indicates it is seen in emission. In the bottom panel, the strength and sign of the averaged signal is shown along with several important landmarks coinciding with the turning points in this curve. The redshift is shown at the top of the bottom panel. The precise timing of the turning points is not well- constrained, this is just one plausible history. As such, the exact redshift values do not completely match those that we described in the text. Figure takenfrom(147)...... 63 1.23 An illustration (Wayne Hu, http://background.uchicago.edu/ whu/) of how a net signal is generated from Thomson scattering due to the presence of a quadrupole . The blue cross and red cross show relatively strong and weak incident radiation, respectively, on an electron at the origin. The red/blue cross indicates the average polarization of scattered light and demonstrates that it obtains a net vertical polarization...... 69

xvi LIST OF FIGURES

1.24 The (simulated) effect of the neutral fraction on the observed clustering of LAEs (taken from McQuinn et al. 108). The top panels show the underly- ing ionization fields, the middle row shows the true location of LAEs in the simulation, and the bottom panel shows the detectable LAEs in the simula- tion. This shows that, LAEs which occupy the same ionized bubble will be observable, resulting in a less homogeneous field of observable LAEs. Each panelis94Mpcacross...... 74 1.25 Several claimed constraints on x during the Epoch of Reionization (mark- h HIi ers), of which we touch on in this section, along with best fit curves calculated using luminosity functions (Robertson et al. 154). The red shaded curve shows the maximum-likelihood model of the neutral fraction (white)

with 1σ errors and is consistent with τe measurements. The analo- gous curve for Robertson et al. (155) is shown in blue, but is in conflict with

the WMAP τe constraints. A model that forces the blue curve to satisfy

the WMAP τe constraint is shown in yellow. This figure demonstrates that, under some assumptions, the scenario where galaxies dominate reionization is not in conflict with the constraints on the timing of the EoR to date. . . 79

2.1 Example mock Ly α forest spectrum and corresponding neutral fraction. The top panel shows the Ly α transmission while the bottom panel is the neutral fraction along the line of sight, with ionized regions set to x 0 for illus- HI ≈ tration. The black curve in the top panel shows the transmission through the forest when absorption due to the hydrogen damping wing is neglected, while the red curve includes damping wing absorption. The comparison illustrates that damping wing absorption has a prominent impact, but it is also clear that the presence of the damping wing will be hard to discern by eye. The line of sight is extracted from a model with x = 0.22, but note that we h HIi have deliberately chosen a sightline with more neutral regions than typical. 94

xvii LIST OF FIGURES

2.2 Dark gap size distribution for the x = 0.22, F = 0.1 model. The solid h HIi h i blue curve shows the total distribution of dark gaps from an ensemble of mock spectra, where the magenta (cyan) curve shows the same thing but for the dark gaps sourced by ionized (neutral) gas. Here, we have focused on dark gaps with L> 0.75 Mpc/h. This clearly demonstrates that neutral hydrogen is the dominant source of large dark gaps in our mock spectra, provided there isanappreciableneutralfraction...... 97 2.3 Large-length tail of the dark gap size histogram for x = 0 (magenta), 0.05 h HIi (cyan), 0.22 (blue), and 0.35 (black) for the case when F = 0.1. The y-axis h i is scaled to indicate the expected number of dark gaps obtainable from 20 spectra. Bins in this figure are spaced logarithmically. The dashed magenta line indicates the dark-gap size distribution in the fully ionized case when the true transmission is F = 0.03, but continuum fitting errors result in a h i measured mean transmission of F = 0.1...... 98 h measi 2.4 Stacking idealized Ly α spectra containing toy HI regions. The above fig- ure shows the stacked transmission outside isolated HI regions with mean density and size L = 0.76 Mpc/h (v 100km/s), L = 1.27 Mpc/h ext ≈ (v 170km/s), and L = 5.34 Mpc/h (v 700km/s) shown in black, ext ≈ ext ≈ blue, and cyan, respectively. The solid red curve shows the stacked trans- mission outside of the same HI regions neglecting the damping wing, which will be the same on average in all cases. In generating these spectra, we as- sume F = 0.1. In this greatly-idealized case, the presence of the hydrogen h i damping wing is seen clearly through extended excess absorption compared to the red curve. Furthermore, we can see that the excess absorption closely follows what we would expect analytically based on multiplying Eq. 2.6 by the overall mean transmission. In this figure, all stacking starts at HI/HII boundaries...... 100

xviii LIST OF FIGURES

2.5 Presence of deuterium absorption revealed through stacking idealized Ly β spectra containing toy neutral regions. The red and black curves show the stacked Ly β transmission redward and blueward, respectively, of toy neutral regions of length L = 5 Mpc/h ( 700km/s) randomly inserted into many ≈ sightlines, with spectra generated assuming F = 0.1. In each case, h Lyαi stacking begins at the underlying HI/HII boundary. We have also mimicked the effect of including foreground Ly α absorption by scaling the feature by the mean transmission in the foreground Ly α forest. This demonstrates that, at least in this idealized case, the presence of deuterium absorption can be easily seen out to 80km/spasttheHI/HIIboundaries...... 104 ∼ 2.6 Ly α stacking results for various neutral fractions. The top panel shows the mean (noiseless) stacked transmission outside of large absorption systems (solid) and small absorption systems (dashed) in the Ly α forest for neutral fractions x = 0.35 (black), 0.22 (blue), 0.05 (red), and 0 (magenta). h HIi The transmission here is estimated from a large ensemble of mock spectra to obtain a smooth estimate of the average transmission around saturated regions in each model. The bottom panel shows the statistical significance of the difference between the dashed and solid curves in the top panel assuming a sample of 20 spectra are used in the stacking process...... 112 2.7 Ly α stacking results assuming F = 0.05. The above panels are identical h i to those in Fig. 2.6 except that mock spectra have been generated assuming F = 0.05...... 113 h i 2.8 Results of Ly α stacking with HIRES-style spectra ( F = 0.1). The above h i panel is identical to the bottom panel in Fig. 2.6 except that the spectra have had the bin size and spectral resolution adjusted to match that of Keck- HIRES spectra. Additionally, we have added noise such that the spectra have a signal-to-noise value of 10 per pixel at the continuum...... 114

xix LIST OF FIGURES

2.9 Deuterium Ly β stacking results for various neutral fractions. The top panel shows the mean ensemble-averaged noiseless stacked transmission moving blueward (solid) and redward (dashed) away from large absorption systems in the Ly β forest for neutral fractions x = 0.35 (black), 0.22 (blue), h HIi 0.05 (cyan), and 0 (magenta). The bottom panel shows the excess blue- ward absorption in units of the standard deviation of the stacked redward transmission,assuming20spectra...... 131 2.10 Results of Ly β stacking with HIRES-style spectra. The above panel is the same as in the bottom panel of Fig. 2.9, except that it is generated using HIRES-style spectra, with spectral resolution of FWHM = 6.7km/s and additive noise with signal to noise of 30 per 2.1 km/s pixel at the continuum. 132 2.11 Mock dark gap size distribution. This figure is identical to Fig. 2.3 except that it uses spectra with spectral resolution FWHM = 100km/s, bin size

∆vbin = 50km/s, and a signal-to-noise ratio of 10 at the continuum. This figure shows the expected histogram of dark gap sizes using 20 spectra with x = 0.35 (black), 0.22 (blue), 0.05 (cyan), and 0 (magenta) at fixed h HIi F = 0.1...... 132 h i

xx LIST OF FIGURES

2.12 Using the Ly β forest to estimate damping-wing-less Ly α transmission. The above figure shows the estimated shape of stacked damping wing absorption for x = 0 (magenta), 0.05 (cyan), 0.22 (blue), and 0.35 (black). The h HIi curves have been normalized to have their mean values peak at 1. Addi- tionally, we show error bars for the fully ionized case and x = 0.35 case h HIi which indicate the scatter in the curves between groups of 20 spectra. The top plot is obtained by using a large ensemble of mock spectra to model a mapping between stacked Ly β transmission and stacked damping-wing-less Ly α transmission and then applying this to groups of 20 spectra. Meanwhile, the bottom figure plots the ratio of the stacked Ly α flux to the stacked Ly β flux, providing a simplified estimate of the damping wing contribution to the absorptionforeachcase...... 133 2.13 Model for the extended damping wing absorption. The left panel shows the components of our model for stacked transmission outside of a neutral re- gion compared to the stacked transmission using mocked spectra (magenta) for x = 0.22. We show the absorption due to the central neutral re- h HIi gion (blue), average absorption due to neighboring, clustered neutral regions (cyan), and the product of the two transmissions (black). These are denoted in the legend as “1-Halo”, “2-Halo”, and “1-Halo + 2-Halo” in analogy with the halo model. In the right-hand panel, we show the comparison between the modelled transmission (dashed) and transmission from stacked mocked spectra (solid) for x = 0.35 (black), 0.22 (blue), and 0.05 (cyan). The h HIi curves in the right-hand figure have been multiplied by the mean transmis- sion (computed here ignoring resonant absorption for illustration). In this appendix, the stacking is done at the HI/HII boundaries and only damping wing absorption is incorporated to demonstrate the extended excess absorp- tion owing to correlated neighboring systems...... 134

xxi LIST OF FIGURES

3.1 The above figure shows the results of stacking Ly α transmission outside of dark gaps in the Ly β portion of the spectrum with L< 300km/s (top) and L > 300km/s (bottom) for dark gaps with 5.5 z 5.7. The solid ≤ gap ≤ curves are generated using mock spectra assuming x = 0 (magenta), h HIi 0.05 (cyan), 0.22 (blue), and 0.35 (black). The dashed green line shows the stacking results for the spectra described in Table 3.1...... 139 3.2 This figure is identical to Fig. 3.1 except we stack outside of dark gaps with 5.7 z 6...... 140 ≤ gap ≤ 3.3 This figure shows the results of stacking Ly β transmission outside of dark gaps with L> 100km/s in the spectra described in Table 3.1. For this figure, we stack outside of dark gaps with 5.5 z 5.7...... 142 ≤ gap ≤ 3.4 This figure is identical to Fig. 3.3 except we stack outside of dark gaps with 5.7 z 6...... 142 ≤ gap ≤ 4.1 Example reionization histories. The red triangles show the simulated volume- average ionization fraction in our semi-numeric High-z reionization model, the black squares are for the Mid-z reionization scenario, and the blue pentagons are for a low redshift (Low-z) reionization model. The black dashed line shows the reionization history computed by solving Eq. 4.1 with ζ = 46, 9 Mmin = 10 M⊙ and C = 3. The semi-numeric efficiency parameters ζ˜(z) in the Mid-z case have been tuned to match this model...... 151

xxii LIST OF FIGURES

4.2 Thermal state of gas elements with a given reionization redshift, as a function of that redshift. In each case, the gas elements are heated to a temperature of T = 2 104 K during reionization, and the residual photo-heating after r × reionization is computed assuming that the (hardened) spectral index of the ionizing sources is α = 1.5 near the HI photoionization edge. Top panel: The

temperature at mean density (T0) for gas elements at each of z = 4.5, 5.0 and 5.5 as a function of their reionization redshift. Bottom panel: This is similar to the top panel, except it shows the slope of the temperature-density relation (γ 1) rather than T . Note that although we assume that gas elements with − 0 a given reionization redshift all land on a well defined temperature-density relation, this will not generally be a good description once we account for the spread in reionization redshift across the universe...... 159 4.3 Reionization redshifts and at z = 5.5 in the low-z reionization model. Left panel: The reionization redshifts for a narrow slice (0.25 Mpc/h thick) through the simulation. Each slice is 130 Mpc/h on a side. The red regions indicate locations with the highest reionization redshifts across the simulation slice, while the dark regions are the last to be reionized. Right panel: The temperature of the same slice as in the top panel. The red areas in this panel show the hottest locations in the slice, and correspond to the dark regions in the top panel that are reionized late. The dark blue regions in the temperature slice, on the other hand, are the coolest regions that reionized first. The color scales are chosen so that 99% of simulation cells in the slice shown here have redshifts and temperatures falling between the minimum andmaximumvaluesonthecolorbar...... 160

xxiii LIST OF FIGURES

4.4 Reionization redshifts and temperatures at z = 5.5 in the high-z reionization model. Identical to Fig. 4.3, except this figure shows the contrasting High-z model. Note that the color scale in this case also encompasses 99% of the reionization redshifts and temperatures in the simulation slice, but that these ranges are different than in the previous figure...... 161 4.5 Temperature density relations at z = 4.5 and z = 5.5 in the Low-z reion- ization model. The blue points show the temperature and density of gas elements from the simulation at z = 5.5, while the black points are the same at z = 4.5. The red short dashed line shows the median simulated temper- ature as a function of density at z = 5.5. The green long dashed line is the same at z = 4.5...... 163 4.6 Temperature density relations at z = 4.5 and z = 5.5 in the High-z reioniza- tion model. Identical to Fig. 4.5, except the results here are for the High-z reionizationmodel...... 164 4.7 Power spectrum of temperature fluctuations in various models. The curves show the power spectrum of δ (x) = (T (x) T )/ T from the simulated T0 0 − h 0i h 0i models. The blue dotted line, the black solid line, and the red short-dashed line are the power spectra at z = 5.5 in the Low-z, Mid-z, and High-z models

respectively. The black long-dashed line shows the δT0 power spectrum at z = 4.5 in the Mid-z model to illustrate how the temperature fluctuations fadewithtime...... 166

xxiv LIST OF FIGURES

4.8 Thermal state at z = 5.5 for various reionization temperature and spectral shape models. This is similar to the z = 5.5 curves in Fig. 4.2, except here we

vary the reionization temperature, Tr, and the spectral shape, α. Increasing

Tr leads to a higher T0 and a flatter γ for recently reionized gas parcels,

while parcels that reionize at sufficiently high redshifts are insensitive to Tr. A harder ionizing spectrum after reionization (smaller α) leads mostly to a

slightly larger value of the asymptotic temperature achieved at high zr. The harder spectrum also slightly hastens the transition of γ to its asymptotic value...... 191 4.9 Temperature density relation at z = 5.5 for various reionization temperatures in the High-z and Low-z models. The “X”s in the legend indicate the color of the points in the corresponding models, while the dashed lines in the same models have different colors to promote visibility. The models in the legend are listed from top to bottom: the highest points and line (indicating the median temperature at various densities) show the T = 2 104 K, Low-z r × model; next is the T = 1 104 K, Low-z model; then the T = 3 104 K, r × r × High-z model; and finally the T = 3 104 K, High-z model...... 192 r × 4.10 Example sightlines and wavelet amplitudes for two different models of the IGM temperature at z 5. The top panel shows δ (x) for an example ∼ F sightlines with T = 2.5 104 K, γ = 1.3 (red dashed) and the same sightline 0 × except with T = 7.5 103 K, γ = 1.3 (black solid). The bottom panel 0 × shows the smoothed wavelet amplitudes, AL, along each spectrum. The lower temperature model has more small scale structure and larger wavelet

amplitudes. The smoothing scale sn = 51 km/s here, while ∆u = 3.2 km/s and L = 1, 000km/s...... 193

xxv LIST OF FIGURES

4.11 Probability distribution of A for various T models at z 5. Each model L 0 ∼ here assumes a perfect temperature density relation with γ = 1.3, and in each case the mean transmitted flux has been fixed – by adjusting the intensity of the ionizing background – to F = 0.20. As in Fig. 4.10, the smoothing h αi scale has been set to sn = 51 km/s, while ∆u = 3.2 km/s and L = 1, 000 km/s...... 194 4.12 Degeneracy with F . Left panel: Although the PDF of A is sensitive to h i L T , this effect is degenerate with the impact of varying F . For instance, 0 h i the model with T = 1.5 104 K and F = 0.20 is closely mimicked by 0 × h i a colder model with T = 7.5 103 K, yet a larger mean transmission of 0 × F = 0.30. Right panel: This illustrates that the degeneracy can be broken h i by measuring the (relatively) large scale flux power spectrum. The curves here show the flux power spectrum, evaluated at a single convenient (larger- scale) wavenumber of k = 0.003 s/km, in each T model as a function of F . 0 h i The triangle and pentagon show the flux power for each model at the F for h i which the wavelet amplitude PDFs are degenerate in the two models. The large scale flux power in these two models differs appreciably and can be used to break the degeneracy. The red dotted and black dotted horizontal lines areintendedonlytoguidetheeye...... 195

xxvi LIST OF FIGURES

4.13 Example sightlines and wavelet amplitudes from the Low-z and High-z reion- ization models. In the models here, the global mean flux is F = 0.1 and h i z = 5.5. In each panel the red dotted line shows a sightline through the T = 3 104 K, Low-z reionization model while the black solid line is the r × same sightline, except in this case the temperature field is drawn from the High-z reionization model (with T = 2 104 K). The simulated density and r × temperature fields have small scale structure added according to the lognor- mal model, as described in the text. Top panel: The simulated temperature

field. Middle panel: The transmission field, δF . Bottom panel: The smoothed

wavelet amplitude with L = 1, 000 km/s, sn = 34 km/s, and ∆u = 2.1 km/s. The transmission fluctuations and wavelet amplitudes are larger than in Fig. 4.10, mostly because of the lower mean transmitted flux adopted here. . . . 196

4.14 Probability distribution of AL for various reionization and temperature mod- els at z = 5.5. Left panel: In this panel all models are normalized to F = 0.2. The solid black curve shows the wavelet amplitudes for the h i High-z reionization model (with T = 2 104 K), while the red dotted and r × blue dashed curves show Low-z reionization models with reionization tem- peratures of T = 2 104 K and T = 3 104 K respectively. The magenta r × r × dot-dashed line shows a homogeneous temperature model for comparison. In this case, the temperature was set to match the median temperature in the Low-z, T = 3 104 K model for gas at the cosmic mean density; the broader r × distribution in the Low-z model reflects the impact of inhomogeneous reion- ization. Right panel: Identical to the top panel, but here the models fix F = 0.1. In each case, the filter scale and pixel size are set to s = 34 km/s h i n and ∆u = 2.1 km/s respectively, while L = 1, 000km/s...... 197

xxvii LIST OF FIGURES

4.15 Heating/cooling rates at z 7. Left panel: The (absolute value of) the rates ∼ for relevant processes in the IGM at T = 104 K as a function of density, assuming that hydrogen is highly ionized and that helium is mostly singly- ionized. Right panel: Similar to the left panel except the rates are shown as a function of temperature for gas at the comic mean density...... 198

5.1 Fourier profile of the Wiener filter, W (k). The filter is averaged over line-of-

sight angle and the results are shown at zfid = 6.9 for simulated models with x = 0.51 (blue dotted), x = 0.68 (cyan dot-dashed), x = 0.79 (green h ii h ii h ii dashed), and x = 0.89(redsolid)...... 211 h ii 5.2 Application of the Wiener filter to simulated data. The results are for our fiducial model with x = 0.79 at z = 6.9. Top-Left: Spatial slice of the h ii fid unfiltered and noise-less 21 cm brightness temperature contrast field (nor-

malized by T0). Top-Right: Simulated signal-to-noise field after applying the Wiener filter to a pure noise field. Bottom-Left: Simulated signal-to- noise field after applying the Wiener filter to the noisy signal. This can be compared with the uncorrupted input signal shown in the top-left panel and the noise realization in the top-right panel. Bottom-Right: Simulated signal- to-noise field after applying the Wiener filter to the noiseless signal. (The filtered noiseless signal shown here is normalized by the standard deviation of the noise to facilitate comparison with the other panels.) All panels show a square section of the MWA field of view transverse to the line of sight with sidelength L = 1 h−1 Gpc . All slice thicknesses are 8 h−1 Mpc . Unless ∼ noted otherwise, the simulation slices in subsequent figures have these same dimensions...... 237

xxviii LIST OF FIGURES

5.3 Impact of foreground cleaning on the Wiener-filtered field. The top slice is a perpendicular, zoomed-in view of the simulated, unfiltered, noise-less brightness temperature contrast. The bottom slice is the signal-to-noise of the same region after applying the Wiener filter to the noisy signal field. The vertical axis shows the line-of-sight direction, with its extent set to the −1 distance scale for foreground removal, Lfg = 185 h Mpc . The horizontal axis shows a dimension transverse to the line of sight and extends 1 h−1 Gpc...... 238 5.4 Expected signal-to-noise ratio at the center of isolated, spherical, ionized bubbles as a function of bubble radius after applying the optimal matched

filter. The curves show the signal-to-noise ratio at zfid = 6.9 for the MWA- 500 at various neutral fractions: x = 0.4 (blue solid), 0.3 (cyan dashed), h HIi and 0.2 (green dot-dashed). For contrast, the red dotted curve indicates the expected signal-to-noise for an interferometer with a field of view and collecting area similar to a 32-tile LOFAR-like antenna array (at x = 0.4)...... 239 h HIi 5.5 Application of the matched filter to simulated data and noise ( x = 0.79 h ii −1 at zfid = 6.9). The template radius of the filter is 35 h Mpc , since this is a commonly detected bubble radius for our matched filter search. Top-Left: Spatial slice of the unfiltered and noise-less 21 cm brightness temperature contrast field. Top-Right: Simulated signal-to-noise field after applying the matched filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the matched filter to the noisy signal. This can be com- pared directly to the top-left panel. Bottom-Right: Simulated signal-to-noise field after applying the matched filter to the noiseless signal. All panels are at the same spatial slice. See text for discussion on interpreting signal-to-noise values...... 240

xxix LIST OF FIGURES

5.6 Impact of foreground cleaning on the matched-filtered field. This is similar to Figure 5.3, except that the results here are for a matched filter with a −1 template radius of RT = 35 h Mpc...... 241 5.7 An example of a detected ionized region. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the detected bubble in the matched-filtered map. Bottom-Left: Detected bubble superimposed on a zoomed-in view of the noise-less unfiltered 21 cm brightness temperature contrast map. Bottom-Right: A perpendicular zoomed-in view of the bubble depicted in the bottom-left panel. All matched- filtered maps use the template radius that minimizes the signal-to-noise at the center of the detected bubble. In the top-left case, the boxlength is L = 1 h−1 Gpc , while in the zoomed-in slices it is L 500 h−1 Mpc . . . . . 242 ≈ 5.8 An example of an ionized region that our algorithm detects as several neigh- boring bubbles. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The main detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the main de- tected bubble in the matched filtered map (solid curve) along with two other nearby detected bubbles (dashed curve). Bottom-Left: The detected bubble superimposed on the zoomed-in, noise-less, unfiltered 21 cm brightness tem- perature contrast map. Again, the additional nearby detected bubbles are shown (dashed curve). Bottom-Right: A perpendicular view of the bubble depicted in the bottom-left panel, with the nearby detected bubbles visible. All matched-filtered maps use the template radius that maximizes the sig- nal to noise at the center of the main detected bubble. The box length in the top-left figure is L = 1 h−1 Gpc , while in the zoomed-in panels, the box length is L = 550 h−1 Mpc...... 243

xxx LIST OF FIGURES

5.9 A measure of the bubble detection success rate. The points ( ) show the × volume-averaged ionized fraction of detected bubbles versus their detected radius. For comparison, the cyan shaded region shows the 1-σ spread in the ionized fraction of randomly placed bubbles of the same radii. The bubble depicted in Fig. 5.7 is marked with a large red square, while the three bubbles shown in Fig. 5.8 are marked with large green circles...... 244 5.10 Size distributions of detected bubbles for varying (volume-averaged) ion- ization fractions. The histograms show the size distribution of (identified) ionized regions for simulation snapshots with volume-averaged ionized frac- tions of x = 0.51 (top-left), 0.68 (top-right), 0.79 (bottom-left), and 0.89 h ii (bottom-right). These figures demonstrate how the total number and size distribution of detected bubbles varies with ionized fraction...... 245 5.11 Bubble detection with the MWA-128. This figure is similar to Figure 5.5, except it is for the MWA-128 configuration rather than for the MWA-500. . 246 5.12 Bubble detection with a LOFAR-style interferometer. This figure is similar to Figure 5.5, except it is for the LOFAR configuration rather than the MWA- 500. Additionally, all boxes in this figure have a side length of 426 h−1 Mpc , corresponding to the field-of-view of the LOFAR-style interferometer at z = 6.9...... 247

xxxi Chapter 1

First Things First

The subject of this thesis is the “Epoch of Reionization”, also known as “hydrogen reion- ization” or “reionization” or simply the “EoR”. Let us start in 1.1 with a brief description § of the reionization process, how it relates to the evolution of the Universe as a whole, and some of the open questions that this thesis aims to address. In 1.2, we will provide a brief § overview of the variety of probes that have been used to constrain reionization to date, while also providing motivation for the development of additional approaches in 1.3. From § there, we will move onto our own work. We begin by discussing several novel methods for utilizing the Ly α and Ly β forest, first to constrain the end of reionization in 2 and 3, and § § then to constrain the temperature of the intergalactic gas at slightly later times in 4. In 5, § § we will continue by developing approaches to directly observe the reionization process with interferometric experiments that will be up and running in the near future. We conclude in 6. §

1.1 Cosmic Context

The problem of reionization arises when trying to reconcile observations of the early Uni- verse with those of the present-day Universe. Namely, Big Bang implies

1 1.1 Cosmic Context

that the early Universe was composed almost entirely of hydrogen and helium and observa- tions of the Cosmic Microwave Background (CMB, discussed in 1.2.3) demonstrate that, § when the Universe was 380,000 years old,1 this hydrogen and helium became neutral. ∼ However, observations of nearby quasars (specifically, the Ly α forest, discussed in 1.2.1) § demonstrate that the gas between the galaxies, dubbed the intergalactic medium (IGM), is almost entirely ionized by at least & 1 billion years after the Big Bang to today (Gunn and Peterson 65).2 Therefore, some process – acting across the entire volume of intergalactic space in the intervening 13 billion years – must have managed to strip the electrons from ∼ almost all atoms, and keep them ionized even today. We refer to this process as the Epoch of Reionization. Technically, several have to take place: ionizing hydrogen, singly ionizing helium, and doubly ionizing helium with the required per ionization being 13.6 eV, 24.6 eV, and 54.4 eV, respectively. Since the first two processes require similar energies, they are thought to occur concurrently. However, the complete ionization of helium requires four times as much as for hydrogen and likely occurs significantly later and as a result of a different process (see, e.g., 6.3.2 of Barkana and Loeb 4). For this thesis, we focus § exclusively on hydrogen ionization, and the single ionization of helium, and treat the term “reionization” as being synonymous with this. To help understand where reionization fits into the evolution of the Universe as a whole, we can refer to Figure 1.1, which shows the qualitative evolution of the Universe starting with inflation and the CMB on the far left and ending with the present-day Universe on the right and spanning 13.8 billion years. One of the exciting aspects of reionization is ∼ 1This is ∼ 3 × 10−3% of its current age. If the Universe were an 80-year-old , then the CMB would provide a picture of him/her when they were less than a day old. 2This was inferred from the lack of Ly α absorption in quasar spectra. However, when this was first observed in 1965, it was not obvious that this was indicative of an ionized IGM rather than a scenario where formation was so efficient as to remove almost all of the gas from the IGM. See Meiksin (115) for a nice review of the physics of the IGM.

2 1.1 Cosmic Context

Figure 1.1: Milestones in the evolution of the Universe from the Big Bang to today.(Photo from NASA WMAP science team)

3 1.1 Cosmic Context

that so little about it is known for sure. However, a reasonable round-number placement of reionization in this figure would be that it was an extended process that occurred somewhere between 200 million years after the Big Bang and 1 billion years after the Big Bang, likely ∼ coinciding with the formation of the first galaxies. The manner in which reionization progressed is also not known for sure and depends on the specifics of what energetic process is at its root. For example, if “softer” sources, emitting ionizing radiation in the UV but not X-ray, drove reionization, then these will see a relatively high photoionization cross section and will not travel far before ionizing hydrogen in their path. Under this scenario, the state of the Universe during reionization would likely resemble a two-phase medium with ionized regions surrounding the ionizing sources and sharp boundaries between the ionized regions and the neutral IGM. Eventually, as the ionized regions grew, they would overlap until the entire volume of the IGM became filled with ionized gas. This is the expected course for reionization in the event that it is driven by the first galaxies. If the ionizing sources mostly have a “harder” spectrum, emitting ionizing radiation strongly in X-ray, then the photoionization cross section of these photons will be relatively low. This is because, for photons with E 13.6eV, the cross section falls off as σ 1/ν3. γ ≫ ∼ Because of this, ionizing radiation from these sources will travel farther on average before ionizing a . This will result in transitions between fully-ionized regions and fully-neutral regions being more gradual. This is the expected scenario in the case where reionization is driven by X-ray binaries or quasars. Figure 1.2 shows examples slices through numerical simulations of reionization that illustrate how it may have proceeded across cosmic time. The ionized regions in the figure (in white) show how ionized “bubbles” form around galaxies and how they grow and merge to fill progressively more of the IGM. Eventually, the entire volume of the IGM is filled with ionized gas. The figure also illustrates one of the ways in which the size of the ionized regions depends on the properties of the ionizing sources. The fraction of the IGM volume that is

4 1.1 Cosmic Context

in the neutral phase (dubbed the neutral fraction, denoted x ) is fixed and decreases for h HIi lower rows. In each column, the ionizing luminosity of a galaxy is assumed to be a power law in the mass of the galaxy’s host dark halo, with the typical host halo mas increasing as one moves from left to right. As such, the rightmost column demonstrates a plausible reionization scenario when very rare and massive sources dominate the ionizing photon budget. We can see this results in larger ionized regions for a fixed neutral fraction and sharper boundaries between the neutral and ionized regions. On the other hand, the left- hand columns correspond to a relatively larger contribution to the ionizing photon budget from less-massive sources. The result is a more homogeneous ionization with the ionized regions being smaller, on average. With this qualitative picture in mind, we briefly discuss the broad ways in which reion- ization is important for our understanding of astrophysics and cosmology. First, the Epoch of Reionization is a significant missing piece in the story of the evolution of the Universe and represents a period in the history of the Universe where we have very few direct ob- servations. Work towards understanding reionization is in line with the overarching goal of pushing observations further and further back in time. Second, the Epoch of Reioniza- tion marks the time when radiation from luminous sources became the dominant influence on the IGM and understanding the source of this radiation is interesting in its own right. Understanding the evolution in the properties and number of these bright sources is essen- tial for complete cosmological models. Additionally, while the best guess for the source of the ionizing radiation is dwarf galaxies, it is possible that reionization studies will reveal more exotic and unexpected scenarios. For example, annihilating or decaying dark mat- ter might play a role in reionizing the Universe (e.g., Kasuya and Kawasaki 77, Mapelli et al. 100, Pierpaoli 144). Third, the temperature and ionization state of the gas in the Universe plays a regulatory role in galaxy formation: hot and ionized gas will take longer to cool and collapse than cold neutral gas. Since reionization significantly affects both the temperature and ionization state of the gas, understanding reionization will be essential for

5 1.1 Cosmic Context

S2 S1 S3 S4 z = 7.7 z = 7.3 z = 8.7

Figure 1.2: Slices through numerical simulations of reionization. The above panels are simu- lation outputs from McQuinn et al. (109) showing four different reionization models. Neutral regions are shown in black and ionized regions are shown in white. Each row is at fixed xHI h i with xHI =0.8 (top), 0.5 (middle), 0.3 (bottom). The luminosities of the ionizing sources are h i related to their mass by N˙ m1/3 (left), N˙ m (left-middle), N˙ m5/3 (right-middle), and ∝ ∝ ∝ N˙ m but with a larger minimum mass (right). Each slice has a sidelength of L = 93 Mpc. ∝

6 1.2 The Shoulders of Giants

understanding subsequent galaxy formation. Consequently, a key goal of modern cosmology is to understand the timing and nature of reionization. When did it happen? How long did it take? What were the ionizing sources? What were the properties of the ionized regions and how did they evolve? These are the questions we aim to examine in this work.

1.2 The Shoulders of Giants

Before we continue, it is first worth appreciating the difficulty of what we are trying to do. Essentially, we care about measuring the properties of the intergalactic gas – not or galaxies – when the Universe was only .1 billion years old, a seemingly impossible task. Fortunately, we are given the invaluable gift that light travels at a finite speed and, as such, if we look at distant objects, we see them as they were in the past. Therefore, if we look at the at the gas between galaxies 13 billion light years away from us, we will see ∼ it as it was roughly 13 billion years ago, when the Universe was only 1 billion years old. This means that, in principle, this information of how the young IGM evolved is directly available to us. However, even taking this into account, the intergalactic gas we care about is not bright and it is located extremely far away, so how are we supposed to observe it? An inspiring aspect of studying the Epoch of Reionization is that – even though it seems impossible to understand the properties of the Universe at such early times – a number of powerful approaches have been developed to determine the nature of reionization. It is this impressive body of work that we aim to build upon. We discuss a selection of the existing and future methods for constraining the EoR in this chapter in order to provide some context and motivation for our work.

1.2.1 The Ly α Forest

Arguably the most powerful tool for constraining the high-redshift IGM to date has been the Ly α forest. This refers to the pattern of absorption lines seen in the spectra of distant

7 1.2 The Shoulders of Giants

bright objects due to intervening hydrogen, as we will discuss. The Ly α forest results, in part, from another invaluable gift to the field of cosmology: the redshifting of light. This redshifting is a consequence of the expansion of the Universe: the wavelengths of photons propagating through the Universe are stretched as the Universe expands. The stretch is seen as a shift in the spectra of distant sources and the precise amount of the shift can be used to infer a cosmological distance to the source via a model for the expansion history of the Universe. Because of this relationship, distances to objects are often measured as a redshift, defined as the fractional increase in wavelength that a photon experiences when travelling from a given distance to us. This is denoted by z and defined according to the expression:

λobserved = λemitted(1 + zemitter). (1.1)

The Ly α forest is seen in the spectrum of extremely bright background objects, usually quasars or gamma-ray burst (GRB) afterglows, after their light has been processed by the intervening gas. Since the intervening gas is primarily composed of hydrogen and since this hydrogen is generally in the ground state, any intervening neutral patches will absorb light from the background object at the Lyman-series wavelengths, with the strongest absorption occurring at the Ly α wavelength: λα = 1216A.˚ If the Universe were not expanding, then all intervening neutral hydrogen would absorb light from the quasar at one wavelength:

λα = 1216A,˚ neglecting the other lines in the Lyman-series for the moment. However, due to the expansion of the Universe, photons emitted from the quasar/GRB blueward of the Ly α line will redshift as they travel towards us. If they encounter neutral hydrogen as they redshift through the Ly α line, then they will be absorbed and an absorption line will be seen in the spectrum of the background quasar at a wavelength blueward of Ly α (in the rest frame of the quasar/GRB). This process is sketched in Figure 1.3. Thus, the Ly α forest is the pattern of absorption lines seen blueward of the rest-frame Ly α line in quasar spectra due to intervening neutral gas.

8 1.2 The Shoulders of Giants

The same logical progression also applies to the other lines in the Lyman-series. There- fore, you could imagine observing a Ly β and Ly γ forest at smaller wavelengths. There are a couple differences, however. First, lines deeper in the series have a smaller cross section for absorption, so intervening hydrogen will absorb less at these frequencies. Second, photons emitted from a background source with energies larger than Ly β will redshift through the Ly β wavelength and also possibly through the Ly α wavelength before reaching us and will have two opportunities to be absorbed. The photon’s physical location when it redshifts through those two wavelengths will be completely different and, therefore, when observing absorption lines in the Ly β forest, it can be difficult to tell if the photons were absorbed by distant gas undergoing a Ly β transmission or closer gas undergoing a Ly α transition. This problem is clearly exacerbated when considering still higher-order lines since a larger number of distinct regions along the line of sight can contribute to the absorption. We show two example quasar spectra in Figure 1.4. The spectrum in the top panel is for a quasar at relatively low redshift and shows very little absorption. Meanwhile, the quasar in the bottom panel shows little absorption for emitted wavelengths redward of Ly α but is heavily punctuated by absorption blueward of Ly α due to intervening neutral hydrogen. At this point, the Ly α forest should sound like a perfect tool: if we want to map the distribution of neutral hydrogen along the line of sight to a distant bright source, we can simply map each absorption line in the Ly α forest to a parcel of neutral hydrogen. However, the story becomes complicated here due to the extremely great tendency for hydrogen atoms to absorb at the Ly α wavelength. The tendency for absorption by a parcel of gas is typically quantified by an “optical depth”, denoted τ. The fraction of light incident on the cloud that emerges unabsorbed, F , is related to the optical depth by

F = e−τ . (1.2)

The optical depth for Ly α absorption of a neutral hydrogen gas parcel is approximately

9 1.2 The Shoulders of Giants

1+ z 3/2 τ 3.3 104x (1 + δ) (1.3) α ≈ × HI 6.5   where x is the fraction of the hydrogen in the cloud that is neutral and δ (ρ HI ≡ − ρ¯)/ρ¯ is the local baryonic overdensity in units of the cosmic mean. We approximate the line profile of the transition as a delta function in frequency here, but we discuss more realistic descriptions of the line profile in 1.2.1.3 and 1.2.1.4. Using this expression, we § § can calculate the minimum neutral fraction needed for a gas parcel at mean density to allow 1% transmission at z = 5.5:

.01 = e−τmin = τ 4.6 (1.4) ⇒ min ≈ 4.6 τ x (1.5) ≈ α HI,min = x 1.4 10−4. (1.6) ⇒ HI,min ≈ × This reveals the fly in the ointment here: even a gas parcel that is 99.9% ionized will allow less than 1% transmission at the redshifts of interest for reionization. Evidently, even highly-ionized gas can lead to near complete absorption in the Ly α line at the redshifts of interest. Therefore, we can not simply map absorption lines in the Ly α forest to regions of significantly-neutral hydrogen. In fact, the second example quasar we see in Figure 1.4 shows significant Ly α absorption and is located at z = 3.62, much later than the end of reionization. The idea that absorption lines in the Ly α forest correspond to isolated parcels of neutral hydrogen is a useful tool in explaining the basic idea here, but is actually quite inaccurate. Instead, it is more accurate to say that absorption in the Ly α forest traces fluctuations in the underlying density field along the line of sight (Croft et al. 39). At the redshifts that we are concerned with, the density of the IGM is such that the forest is significantly more absorbed than shown in Figure 1.4, with isolated absorption lines becoming exceedingly rare.

10 1.2 The Shoulders of Giants

At this point, the reader may ask what utility does the Ly α forest have at all? Well, an enormous amount. With the fluctuating pattern of transmission and absorption in the Ly α forest in part tracing line-of-sight fluctuations in the underlying matter distribution, we are able to use it to constrain the , measure acoustic oscillations, and put lower limits on the mass of the (Viel et al. 178), for example. Additionally, absorption features due to damped Ly α absorbers (DLAs) can be used to measure the primordial deuterium abundance as a test of . suspense! But how to constrain the EoR?

11 1.2 The Shoulders of Giants

Figure 1.3: Illustration of the basic physics behind the Ly α forest and how gas at different locations along the line of sight results in absorption lines at different wavelengths. (Image from http://www.astro.ucla.edu/)

12 1.2 The Shoulders of Giants

Figure 1.4: Flux as a function of rest-frame wavelength for a quasar at z = 0.158 (top) and z = 3.62 (bottom). The denser IGM at higher z results in a dense “forest” of ab- sorption lines blueward of the rest-frame Ly α line (1216A)˚ in the lower panel. (Image from http://www.astro.ucla.edu/)

13 1.2 The Shoulders of Giants

1.2.1.1 Evolution of τeff

Perhaps the most common analysis performed on high-redshift quasar spectra in the context of constraining the EoR is measurements of the effective Gunn-Peterson optical depth, defined as

F e−τeff (1.7) h i≡ where F is the averaged transmission fraction over a redshift bin in a quasar/GRB spec- h i trum. Under the assumption of a uniform ionizing background and ionization equilibrium, where the rate that neutral hydrogen atoms are ionized is equal to the rate that ionized hydrogen atoms recombine, the effective optical depth encodes important information about the state of the IGM. In order to see this, we can take a few steps to express the optical depth in terms of the properties of the IGM.1 First, the Gunn-Peterson optical depth can be expressed as

2 πe nHI τGP = fαλα , (1.8) mec H(z)

where H(z) is the Hubble parameter at redshift z, e is the charge of the electron, me

is the electron mass, c is the speed of light, λα is the Ly α wavelength, fα is the quantum mechanical oscillator strength, and nHI is the number density of neutral hydrogen atoms. All of these quantities are known with the exception of the number density of neutral hydrogen atoms. To find this, we first utilize the statement of ionization equilibrium:

1The following discussion will borrow heavily from Faucher-Giguere et al. (53) and Fan et al. (49).

14 1.2 The Shoulders of Giants

ΓHInHI = R(T )nenHII (1.9)

R(T )nenHII nHI = (1.10) ΓHI R(T )ne xHI = (1.11) ΓHI

where ΓHI is the photoionization rate due to the ionizing sources, ne is the number density of free electrons, nHII is the number density of ionized hydrogen atoms (),

and xHI is the hydrogen neutral fraction. The left-hand side of Eq. 1.9 represents the rate of photoionizations per volume and the right hand side represents the rate of hydrogen recombinations per volume. Under the assumption of ionization equilibrium with a uniform ionizing background, the presence of any transmission suggests n n + n = n and HII ≈ HI HII H

2 ρc(z)Ωb(z)(1 YHe) 3H (z) Ωb(z)(1 YHe) n¯H = − = − (1.12) mp 8πG mp 3H2Ω (1 Y ) = 0 b,0 − He (1 + z)3 (1.13) 8πG mp

nH = (1+ δ)¯nH. (1.14)

In this expression, ρc is the critical density for a flat Universe, Ωb is the baryon density

in units of the critical density, YHe is the fraction of baryonic mass in the form of helium such that (1 Y ) is the fraction of baryonic mass in the form of hydrogen, and m is the − He p mass of the proton which is effectively equal to the mass of the hydrogen atom. A subscript

of “0” denotes that these are present-day values andn ¯H denotes the average of nH. Thus, as we expect, this expression is essentially equal to the mass density of hydrogen atoms in the Universe divided by the mass per atom.1 The expression for the electron number

1It may be interesting to note that this value corresponds to 0.2 hydrogen atoms per cubic meter today and roughly ∼50 hydrogen atoms per cubic meter at z = 5.5. It is very empty out there.

15 1.2 The Shoulders of Giants

density should be the same, since each ionized hydrogen atom releases one free electron. However, provided helium is singly-ionized along with hydrogen, the number density will increase according to:

3H2 (1 Y ) Y n¯ =n ¯ +n ¯ = − He + He (1.15) e H He 8πG m 4m  p p  1.08¯n . (1.16) ≈ H For simplicity in this discussion, but not in the body of this thesis, let us approximate n n n . The quantity R(T ) in Eq. 1.9 is the recombination rate, which is equal to tot ≡ e ≈ H (Hui and Gnedin 72):

T −0.7 R(T ) 4.2 10−13 cm3 sec−1 . (1.17) ≈ × 104K   For δ . 5, Hui and Gnedin (72) showed that the temperature and density follow the relationship

T T (1 + δ)γ−1 (1.18) ≈ 0

where T0 is the temperature of a parcel of gas at mean density, and γ is the slope of the temperature-density relation. This tight relationship will become less accurate as one approaches reionization, as we discuss in 4. For compactness, let’s define R R(T = § 4 ≡ 104K). At this point, we are ready to combine Eq. 1.18, 1.17, 1.14, 1.10, and 1.8 to get an expression for τGP:

2 −0.7(γ−1) 2 2 πe fαλα R4(1 + δ) n¯tot(z)(1 + δ) τGP = (1.19) mec H(z) ΓHI πe2 f λ R n¯2 (z) = α α 4 tot (1 + δ)2−0.7(γ−1). (1.20) mec H(z) ΓHI

16 1.2 The Shoulders of Giants

Finally, we have an expression for the Ly α optical depth in terms of several properties of the IGM. The primary unknown in the above expression is the photoionization rate, which is a very complicated parameter that depends on the number, intensity, spectrum, and prox- imity of ionizing sources among other things. In most previous work, the photoionization rate has been approximated as spatially uniform. This approximation is well-motivated at z 5 or so, when the mean free path to ionization photons is inferred to be rather ≤ long (e.g., Prochaska et al. 149, Worseck et al. 183). At these redshifts, each gas parcel is exposed to ionizing radiation from many sources and so fluctuations in the radiation field are correspondingly small. During reionization, however, the photoionization rate will have large spatial fluctuations: there will be neutral regions that have not yet been exposed to radiation, and even the radiation field incident on ionized parcels will vary with the size of the ionized region that the parcel belongs to, and on other IGM properties. Regardless, with this approximation, the observed mean transmission in a region of the spectrum is akin to an average of Eq. 1.20 marginalizing over the density field:

F = dδ e−τ(δ)P (δ) e−τeff , (1.21) h i ≡ Z where we substitute our expression in Eq. 1.20 for τ(δ) above. It is then interesting to adopt a model for the probability distribution of the underlying density field (extracted from numerical simulations of cosmological ), and determine the value of Γ (assumed to be uniform) that matches the observed mean flux, F . With estimates HI h i of the photoionization rate in hand, we can utilize Eq. 1.11 in order to obtain measurements of the IGM neutral fraction in each redshift bin, thus providing us with a handle on the progress of the EoR. Results for measurements of Γ and x via this method, performed HI h HIi by Fan et al. (50), are shown in Figure 1.5. This figure demonstrates that, using τeff and the assumption of ionization equilibrium with a uniform ΓHI, estimates of the neutral fraction are exceedingly small for z . 6. This argument has played a large part in forming the common knowledge that reionization has ended by z = 6.

17 1.2 The Shoulders of Giants

Figure 1.5: The inferred evolution of the photoionization rate, ΓHI (left), and neutral fraction (right) from Fan et al. (50). In the left-hand panel, measurements of the effective optical depth in the Ly α (blue), Ly β (green), and Lyγ (magenta) forest are converted to estimates of the photoionization rate, with arrows indicating upper bounds. The small circles are measurements in individual redshift bins over the 19 quasars used with the large circles being averages. In the right-hand panel, measurements of the photoionization rate are converted to estimates of the volume-averaged neutral fraction.

18 1.2 The Shoulders of Giants

Despite the widespread analysis of τeff in constraining the end of reionization, the inter- 1 pretation of τeff is quite complicated. In particular, assuming that the ionization state of the IGM is determined by a spatially-uniform ΓHI is tantamount to assuming that reioniza- tion has, in fact, completed. Specifically, as discussed in 1.1, reionization is likely a highly § inhomogeneous process with ionized bubbles forming around the brightest sources, growing, and eventually overlapping. During the period prior to complete overlap, regions of neutral hydrogen will be shielded from the ionizing radiation while ionized bubbles will experience

a very large ΓHI. This is not reflected in Eq. 1.9 and so we expect conclusions derived from this method to be unreliable when we begin to push up against the end of reionization.

1For a more thorough discussion of controversial aspects of these constraints, see the intro to McGreer et al. (105).

19 1.2 The Shoulders of Giants

1.2.1.2 Dark Pixel Covering Fraction

As demonstrated in the previous section, interpreting measurements of the effective optical depth in the context of reionization is complicated and can rely on controversial assumptions. However, an alternative approach is to consider what constraints can be made without resorting to such assumptions. In this regard, measurements of the dark pixel covering fraction in high redshift quasars can be used to place robust upper limits on the fraction of the IGM volume that is in the neutral phase, x . This approach is rooted in the fact h HIi that neutral parcels of gas are certain to result in saturated absorption in quasar spectra 4 due to their optical depths being τHI & 10 (Eq. 1.8). Therefore, a reliable upper bound on the neutral fraction at a given redshift can be estimated by the fraction of pixels in quasar spectra that are completely absorbed at that redshift. An obvious drawback of this method is that, at z 6, overdense yet ionized regions will ∼ also result in saturated absorption and may significantly increase this upper bound on the neutral fraction. One approach to combat this effect is to incorporate the Ly β forest into the analysis. The optical depth for Lyman-series transitions scales as fλ, where f is the oscillator strength of the transition and λ is the corresponding wavelength. Therefore, the analogous expression of Eq. 1.8 for Ly β is:

f λ 1+ z 3/2 τ = τ β β 5.3 103x (1 + δ) . (1.22) β α × f λ ≈ × HI 6.5 α α  

where fα = 0.4162, λα = 1216A,˚ fβ = 0.0791, and λβ = 1026A.˚ From this expression, we can see that a mean-density parcel of neutral gas should cause saturated absorption in both the Ly α and Ly β transitions. Meanwhile, ionized overdense regions are less likely to cause saturated absorption as their optical depth in Ly β is reduced by a factor of f λ /f λ β β α α ≈ 1/6. Therefore, limits from the dark-pixel covering fraction may be improved by requiring simultaneous absorption in both Ly α and Ly β as part of the definition of a dark pixel. Additionally, the Ly β dark pixel covering fraction on its own is a viable tool for establishing

20 1.2 The Shoulders of Giants

an upper bound on the neutral fraction, although foreground Ly α absorption may undo some of the gains from the lower τβ value. In practice, all three approaches (requiring Ly α, Ly β, and Ly α+Ly β absorption) are used. This procedure faces several complications when actually carried out, however. First, there are several sources of random noise that add scatter to each observed quasar spectrum. This can result in spurious transmission in pixels that otherwise would have been completely absorbed. Therefore, to measure the dark pixel fraction in quasar spectra, one first needs to create a suitable definition of what qualifies as a “dark” pixel. One approach here is to define dark pixels as having transmission below some threshold defined in terms of the noise standard deviation, σN. This presents us with a trade-off, however, since larger thresholds will reduce the number of neutral pixels we miss but also increase the number of ionized pixels that get incorporated into the dark pixel population. Alternatively – provided the noise has zero median – half of all truly-absorbed pixels will result in negative flux values, on average. This presents the possibility of using twice the negative-flux pixel covering fraction as an estimate of the dark-pixel covering fraction (or four times, in the case of requiring Ly α+Ly β absorption) (McGreer et al. 104). A second complication is that, since pixels have a finite width, their transmission val- ues effectively represent an average of the transmission over some region in the spectra. If the pixel width is large enough, then it is possible for a pixel to have non-zero trans- mission despite corresponding to a physical region that contains significantly-neutral gas. For example, if the physical region in space associated with the pixel is 80% composed of completely-neutral gas and 20% composed of completely ionized gas which allows full trans- mission, then the transmission of that pixel will be 20% and will likely not qualify as a “dark pixel” despite containing neutral gas. Thus, even in a measurement as seemingly-simple as the dark-pixel covering fraction, these details must be kept in mind when interpreting results.

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Regardless, McGreer et al. (105) and McGreer et al. (104) apply the dark-pixel covering fraction approach to 22 high-redshift quasar spectra to produce the constraints on x h HIi shown in Figure 1.6. Dimly-colored points correspond to McGreer et al. (105) while bold- colored points correspond to McGreer et al. (104). These results present a very different interpretation than using τeff measurements while using the same data. Namely, this model- independent analysis does not in fact require reionization to complete by z . 6, contrary to much of the conventional wisdom in the reionization field.

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1.0

0.8

0.6 HI ¯ x 0.4

0.2

0.0 5.2 5.4 5.6 5.8 6.0 z

Figure 1.6: Current limits on xHI derived from the dark-pixel covering fraction in McGreer h i et al. (104). Lightly-shaded points are older limits obtained in McGreer et al. (105).

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1.2.1.3 Damping Wing Redward of Ly α

Much of the difficulties in using the Ly α forest to constrain the timing of the EoR can be boiled down to the following problem: interpreting Ly α absorption in high-redshift quasar spectra is difficult because both neutral and ionized gas can result in saturated absorption. Therefore, it is worth asking if there are any ways to break this degeneracy in Ly α absorption in order to determine which absorption is likely due to neutral hydrogen. One potential approach toward this goal, which has received much attention (e.g., Miralda- Escude 122, Chornock et al. 34, Chornock et al. 33, Mortlock et al. 129, Bolton et al. 21), is looking for the hydrogen damping wing redward of the Ly α line. To understand this approach, let us first understand what the hydrogen damping wing is. For many applications, it is suitable to consider an atom’s ability to absorb radiation as a series of delta functions in frequency: when incident radiation has a frequency exactly coinciding with the energy of the transition, then there is a non-zero probability for ab- sorption and zero probability otherwise. In reality, the probability of absorbing a photon of a given frequency, i.e., the line profile, is a continuous distribution which, while small for frequencies ν = ν , is non-zero. 6 0 The intrinsic line profile for the Ly α transition in the hydrogen atom can be seen as arising from the time/energy uncertainty principle, ∆E ∆t & ~. Specifically, the finite · lifetime of the n = 2 implies the existence of a range of energies that can excite, or result from, the transition. The distribution of this range of energies follows a Lorentzian distribution:1

1 Γ/4π2 φ(ν)= (1.23) π (ν ν )2 + (Γ/4π)2 − 0 with the corresponding absorption cross section

1A quantum-mechanical discussion of this result can be found in §5.8 of Sakurai and Napolitano (158). A classical derivation can be found in §3.6 of Rybicki and Lightman (157).

24 1.2 The Shoulders of Giants

πe2 σα(ν)= fαφ(ν), (1.24) mec where Γ is the decay rate of the transition. The “damping wing” refers to the σ ∼ 1/(ν ν )2 behavior far from line center. This can be used to break the degeneracy between − 0 HII absorption and HI absorption because the optical depth is so much smaller in the damping wing that, without significantly-neutral gas (optical depth scales with neutral fraction), the optical depth at such frequency separations will not be sufficient to cause absorption. Furthermore, the damping wing has a distinct shape which can be fit for in order to infer the properties of the neutral gas which sources it. For example, in an extended neutral region, the damping wings from the different parts of the cloud will add together to slow the decay of the overall absorption. As such, compact absorbers will have a narrower damping wing than extended absorbers. This aids us in distinguishing absorption owing to neutral hydrogen in the diffuse IGM from that due to compact DLAs. While the damping wing from an isolated neutral region in a sea of fully-ionized, τ = 0 hydrogen would stand out like a sore thumb, in reality absorption from the surrounding dense, yet ionized, gas will punctuate the damping wing with additional absorption features and will make it harder to detect. This makes the prospect of looking for isolated damping wings in typical regions in quasar spectra unappealing. However, photons emitted slightly redward of Ly α cannot be absorbed by dense ionized gas since ionized gas has a negligible optical depth for ν = ν . Neutral hydrogen, on the other hand, will allow absorption to take 6 α place redward of Ly α due to the significant optical depth in the damping wing. Because of this, searches for the damping wing slightly redward of the Ly α line will be able to avoid nuisance absorption from neighboring ionized gas. We show a famous example of a potential damping-wing detection in Figure 1.7, taken from Mortlock et al. (129). This shows a region of the transmission spectrum for a quasar at redshift z = 7.084 (ULAS J1120+0641). The fractional transmission nearby the Ly α line

25 1.2 The Shoulders of Giants

exhibits a gradual recovery from almost complete absorption at λ < λα to almost complete transmission at λ > λα, occurring over a wavelength interval consistent with a hydrogen damping wing. The curves in blue show models for damping wing absorption associated with an IGM with neutral fraction x = 0.1 (top), 0.5 (middle), and 1 (bottom) with h HIi a sharp ionization front at a distance of 2.2Mpc from the quasar. In green, a model for the absorption profile of a Damped Ly α Absorber (DLA, see glossary for definition) with column density N = 4 1020cm−2 located 2.6 Mpc from the quasar is shown. Thus, the HI × transmission profile appears consistent with both a significantly-neutral ( x > 0.1) IGM h HIi or a proximate DLA. However, Simcoe et al. (167) perform a search for metal lines, which typically accompany DLA absorption, and find that the gas is extremely metal-poor. This bolsters the claim that the damping-wing absorption seen in this example is, in fact, due to diffuse neutral hydrogen in the IGM. Other searches for damping-wing absorption redward of Ly α have been carried out on, for example, GRB 130606A (Chornock et al. 34) and GRB 140515A (Chornock et al. 33). These authors looked for the damping wing in the spectra of GRB afterglows at redshift z = 5.913 and z = 6.33, respectively. A non-detection in the spectra of the z = 5.913 GRB allowed the authors to place a 2σ limit on the nearby IGM neutral fraction of x < 0.11. h HIi Similarly, no strong evidence of a damping wing was found in the spectrum of GRB 140515A, shown in Figure 1.8. The right-hand panel shows the transmission fraction nearby the Ly α 18.62 −2 transition, which is equally-well fit by pure host absorption (blue, NHI = 10 cm ), pure IGM absorption from gas at 6.0 z 6.328 with x = 0.056 (red), and a hybrid model ≤ ≤ h HIi with a host absorber lying within an ionized bubble with R = 10 comoving Mpc met by an IGM with x = 0.12 (green). As such, they argue against a significantly-neutral IGM at h HIi this redshift. It is worth pointing out, however, that the method of searching for the damping wing redward of Ly α is not without drawbacks. First, detecting the damping wing redward of Ly α relies on your ability to understand what the quasar flux would have been in the

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Figure 1.7: Quasar ULAS J1120+0641 identified at redshift z =7.085 along with several fits for the damping wing.

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absence of the absorbing gas nearby the Ly α line (this unabsorbed flux is referred to as the quasar continuum and predicting the unabsorbed flux for a given quasar is called continuum fitting). Predicting the Ly α line properties in quasars is notoriously complicated and so modelling the precise fractional transmission must be done with care. Second, searching for the damping wing redward of Ly α inherently involves measuring the gas properties nearby the quasar. However, quasars are extremely rare and special objects and it is not obvious that their surroundings are representative of the IGM on average. For example, Lidz et al. (88) found that quasars are likely born into large galaxy-generated ionized regions, suggesting that interpreting the lack of a damping wing detection is not straightforward. Gamma-ray burst afterglow spectra are gaining attention in this regard (See Salvaterra 159 for a review) as they tend to occupy more typical regions of space and have an easier-to- model continuum flux. The drawbacks of GRBs, though, is that they are often accompanied by a host absorber whose damping-wing absorption must be separated from that of the IGM. Third, even when provided with a clean detection of the damping wing redward of Ly α, this will only tell you about one region of space and it will be difficult to use this single observation to extrapolate to the ionization state of the IGM as a whole. Later in this work, we propose a technique for searching for the hydrogen damping wing which, while faced with its own difficulties, is able to avoid the difficulties mentioned above.

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Redshift of Lyα Rest Wavelength (Å) ) 4.5 5.0 5.5 6.0 6.5 7.0 1210 1220 1230 −1 4 Å ⊕ 1.0 −1 s

−2 3 0.8 GRB 140515A 2 at z=6.327 0.6 erg cm Host only 0.4 −17 1 C IV(z=4.804) Al II IGM only Combined 0.2 , 10 λ

with R =10 Mpc Transmission Fraction 0 b 0.0

Flux (f 7000 8000 9000 10000 8900 9000 Observed Wavelength (Å) Observed Wavelength (Å)

Figure 1.8: Spectrum of GRB140515A, a gamma-ray burst located at z = 6.33. The right- hand panel overlays damping wing models from a host absorber (blue), a pure IGM model with

xHI =0.056 (red), and a combination model (green). The authors argue that, while each curve h i provides an equally-good fit to the data, the sharp rise in transmission shown is inconsistent with a significantly-neutral IGM.

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1.2.1.4 IGM Temperature

A detection of a damping wing redward of the Ly α line in a quasar spectra would con- stitute a “smoking gun” for significantly-neutral regions in the IGM, provided you could rule out the possibility of a DLA source. However, in the absence of a smoking gun, a warm gun could be an indication of reionization having completed recently. Specifically, measurements of the IGM temperature can provide us with additional insights about the process of reionization. The utility of the IGM temperature in studying reionization stems mostly from the long cooling time of the low-density gas. The gradual cooling of the gas implies that it retains some memory of when and how it was ionized. Typically, the gas is photo-heated to temperatures of 20, 000K during reionization. The main cooling mech- ∼ anism is adiabatic cooling from the expansion of the Universe, although Compton cooling off of the CMB is important for gas that is ionized at sufficiently early times, at z & 10 or so. In general, gas that was ionized early on will have longer to cool and will reach a lower temperature sooner than gas that ionized more recently. Since the memory of prior photo-heating gradually fades, this measurement is most powerful if it can be made as close as possible to reionization (e.g., Miralda-Escud´eand Rees 124, Hui and Gnedin 73, Lidz and Malloy 87). Because of this relatively simple cooling behavior, it should be possible to turn a measurement of the temperature of the IGM into a constraint on the timing of reionization. In order to do this, we need two main ingredients: a method of measuring the temperature of high-redshift gas and an understanding of how the temperature of the gas evolves with time after being ionized. One popular method for determining the temperature of the IGM utilizes the width of absorption lines in the Ly α forest. In 1.2.1.3, we described the line profile for Ly α § absorption as obeying a Lorentzian distribution. While this is technically correct for any given atom, in reality, the atoms themselves have random thermal motions according to their temperature and will therefore see incident radiation as being redshifted or blueshifted

30 1.2 The Shoulders of Giants

accordingly.1 As such, a hydrogen atom travelling away a photon with frequency just above the Ly α frequency will see the light redshifted and can increase the chance of absorption. The effect of this is that the line profile for Ly α absorption from a gas parcel gets smeared out, or, more precisely, gets convolved the with Maxwell-Boltzmann distribution. The greater the temperature, the greater the extent of this smearing. The Maxwell-Boltzmann distribution describes the velocities of in an ideal gas with a given temperature:

1/2 mp 2 W (ξ)dξ = e−mpξ /2kB T dξ (1.25) 2πk T  Bs  −1/2 2 2 2 −ξ /ξ0 = πξ0 e dξ (1.26)

2kBT ξ0 (1.27) ≡ s mp

where T is the temperature of the gas, ξ is the velocity and kB is the Boltzmann constant. Incident radiation will appear red/blueshifted in the frame of the absorbing atom with the shift being proportional to the atom’s velocity parallel to the incident radiation, as shown in Figure 1.9. Specifically, a photon with frequency ν will be observed by the atom to have frequency ν(1 ξ/c), where ξ is the component of the atom’s velocity away from − the incident radiation. Convolving our line profile with a Maxwell-Boltzmann distribution effectively involves replacing our expression in Eq. 1.24 with

πe2 ∞ σ f dξ φ(ν(1 ξ/c))W (ξ) (1.28) → m c α − e Z−∞ 2 ∞ πe −1/2 2 2 = f dξ φ(ν(1 ξ/c)) πξ2 e−ξ /ξ0 (1.29) m c α − 0 e Z−∞ 2 ∞  −ξ2/ξ2 πe f −1/2 (Γ/4π)e 0 = α πξ2 dξ . (1.30) m c π 0 (ν ν (1 ξ/c))2 + (Γ/4π)2 e Z−∞ − 0 −  1The following discussion of Doppler broadening and the derivation of the Voigt profile closely follows notes taken from Masao Sako’s “Radiative Transfer” class offered in the Spring of 2010.

31 1.2 The Shoulders of Giants

Here, we can make a couple substitutions and redefinitions:

ξ ∆v ν 0 y ξ/ξ (1.31) D ≡ 0 c ≡− 0 ν ν Γ v − 0 a . (1.32) ≡ ∆vD ≡ 4π∆vD

The quantity ∆vD is known as the “Doppler parameter” and is the red/ in frequency space that the atom sees due to its thermal motion. The quantity v is just the distance from line center in velocity space in units of the Doppler parameter. The quantity a (Γ/4π)/∆v represents the ratio of the natural line width to the Doppler ≡ D width. Rewriting our expression in Eq. 1.30, we obtain

2 πe2 1 a ∞ e−y σ(ν)= f dy (1.33) m c α √π∆v π (v y)2 + a2 e D Z−∞ − √πe2 H(a, v) fα (1.34) ≡ mec ∆vD where H(a, v) is known as the Hjerting function or the Voigt function. So finally, we have obtained an expression for the Ly α line profile incorporating the natural line width and Doppler broadening. Overall, the effect of the temperature acts to smear out the absorption line on scales of order 10km/s from line center. The greater the temperature, the larger ∼ the Doppler parameter and the greater the extent of the smearing. At scales & 100km/s, the damping wing dominates the line profile. The profile is complex, but nonetheless, it encodes information about the underlying IGM temperature. Therefore, comparisons between Voigt profiles from actual spectra and mock spectra constructed from cosmological simulations should provide insight into the thermal properties of the IGM. This is the procedure undertaken by Bolton et al. (18). In Figure 1.10, they look for Ly α absorption lines in the proximity zones of a high-redshift quasar in order to fit for the associated Doppler parameter and make inferences about the temperature. The top panel

32 1.2 The Shoulders of Giants

shows the fractional transmission for a mock quasar spectrum. The dashed curves indicate regions where absorption-line fitting was carried out and the vertical arrows indicate where the line centers were found from the fits. The second and third panel show the underlying temperature and density field, respectively. The bottom panel shows the true spectrum in question along with the same information regarding the line profile fits. These authors were able to use these fits to make inferences about the temperature of the IGM nearby the quasar. From these temperature measurements, and through comparison to mock spectra properties, these authors were able to place interesting limits on the ending of reionization

(zH < 9 assuming that the quasar reionizes its vicinity and Pop II stars drive reionization). This specific measurement is very difficult, however. The argument is essentially that the inferred temperature of the gas is too hot for reionization to have ended long before z = 9, otherwise the gas would have had more time to cool below the measured temperature. However, the regions nearby quasars should see a significantly-enhanced ionization field and all constraints made from measurements within this region hinge on ones ability to accurately account for such effects. Although the main drawback of the previous method is that it works only in a limited stretch of spectrum close to the quasar itself, a second disadvantage is that it requires fitting discrete absorption lines to the forest. Decomposing the Ly α forest into a set of discrete absorption lines is not so well-motivated. The forest is better viewed as a continuously fluctuating field that traces underlying fluctuations in the line of sight density field (Croft et al. 39). This is especially true in the high-redshift regions of interest, z & 5, in which essentially all regions of the forest show some absorption. In this regime, the forest is “inverted” in that, instead of stretches of transmission punctuated with absorption features, there are regions of heavy absorption with some transmissive segments interspersed. This renders the goal of fitting for individual Voigt profiles impossible in the IGM. In fact, in order to apply this technique, Bolton et al. (18) analyzed gas in quasar proximity zones,

33 1.2 The Shoulders of Giants

the region surrounding a quasar where transmission is significantly enhanced due to the quasar’s extra ionizing radiation. An alternative approach to fitting line profiles is to measure the small scale power of the flux fluctuations (e.g., Lidz et al. 85, Theuns et al. 172, Zaldarriaga 193). This involves applying a localized wavelet filter to measurements of the Ly α forest in order to measure the level of small-scale fluctuations as a function of position across each spectrum. As we mentioned, large temperatures will lead to a larger Doppler parameter which will smear out small-scale structure. Thus, a large response to a small-scale wavelet filter would indicate significant small-scale structure and suggest a lower temperature for the gas. This approach has the important advantage that it doesn’t rely on individual absorption lines to be discernible in order to extract temperature measurements. In 4, we show that this can § be used to constrain the IGM temperature at z > 5, even in typical regions of the IGM.

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Figure 1.9: Schematic representation of Doppler broadening. The HI atom is moving away with velocity v from incoming radiation with frequency ν. The observed frequency of the radiation in the atom’s rest frame is ν(1 ξ/c) where ξ is the component of the velocity parallel − with the incident radiation.

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Figure 1.10: Measuring the temperature of the IGM in z & 6 quasar proximity zones. This figure shows mock spectra, and corresponding simulated IGM properties, from Bolton et al. (18) in the top four panels. The bottom panel shows the observed spectrum from SDSS J0818+1722, which Bolton et al. (18) use in order to make temperature measurements inside the proximity zone. Dashed lines indicate regions where Voigt-profile fitting was performed and downward arrows indicate the detected centers of the Voigt profiles.

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1.2.2 The 21-cm Line

The 21-cm line refers to the hyperfine splitting of the hydrogen atom where, due to the interaction between the magnetic dipole moments of the electron and proton, a small energy difference exists between the configuration where the spins of the electron and proton are aligned versus where they are anti-aligned. The configuration where the spins are anti- aligned (and magnetic dipole moments are therefore aligned) is energetically favored and has a lower energy by ∆E = 6 10−6eV.1 Thus, -flip transitions will result in (from) × the emission (absorption) of a photon with λ = 21cm, ν = 1420 MHz. This is shown schematically in Figure 1.11. As we discussed in 1.2.1, the cross section or the Ly α transition is extremely large, § which presents us with a host of difficulties. However, the Universe is essentially transparent to 21-cm photons, allowing them to travel unimpeded from distant neutral hydrogen to us. In principal, this allows us to observe the hydrogen density field directly up to redshifts of z 150, far beyond the timescale of reionization and far beyond the reach of the Ly α ∼ forest. In principal, detailed measurements of the 21-cm signal as a function of redshift and angle on the sky should provide a “movie” of the reionization process and finally reveal the nature of the EoR in its entirety. Mapping the intensity of the 21-cm line during reionization avoids many other drawbacks of the Ly α line as well. Namely, would provide us with a 3D volume of intensity values, as opposed to the Ly α forest which is typically observed one sightline at a time. As we will discuss, the 21-cm line is a weak transition and so the emission from fully-neutral regions in the IGM is unsaturated, unlike the case of the much stronger Ly α line. This should greatly facilitate the interpretation of 21-cm measurements. In this section, let us start with an overview of the physics of the 21-cm line and then continue by discussing a couple avenues by which the 21-cm signal could be utilized in order to constrain reionization.

1This is in contrast to a Ly α photon which has energy E = 10.2eV, more than a million times greater.

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1.2.2.1 The Intensity of the 21-cm Line

Let us start by considering the brightness of 21-cm emission from a distant hydrogen gas cloud.1 This is usually described in terms of a specific intensity and then in terms of a brightness temperature. The specific intensity of light leaving our HI cloud is the amount of energy per unit frequency, time, area, and solid angle, denoted Iν . The brightness temperature is the required temperature of a blackbody to radiate with the same specific

intensity at that frequency, i.e., Bν(Tb)= Iν, where Bν denotes the blackbody spectrum. The light we observe from our HI cloud will be a combination of background light shining through the cloud and light emitted by the cloud itself. This follows the radiative transfer equation (in the clouds frame)

T (ν)= T (1 e−τν )+ T e−τν (1.35) b cloud − background where the background light is from the CMB, such that Tbackground = 2.73K(1 + z), and

Tcloud is the spin temperature of the cloud, defined below. The quantity τν is the frequency- dependent optical depth for absorption by the HI cloud. This is equal to

τ = dsσ 1 e−E10/kBTS φ(ν)n (1.36) ν 01 − 0 Z   where the integral is carried out along the line of sight through the cloud. In this expression, E = 6 10−6eV is the energy of the transition, φ(ν) is the line profile, σ is 10 × 01 the cross-section for 21-cm absorption by a hydrogen atom and n0 is the number density of hydrogen atoms in the unexcited state. We follow the convention of Furlanetto et al. (59) and denote the lower energy level by “0” and the higher energy level by “1”. The ratio

1This discussion will closely follow §2.1 of Furlanetto et al. 59, which is an excellent review of the 21-cm line.

38 1.2 The Shoulders of Giants

of the population of atoms in the excited state to the ground state is defined by the spin temperature and the degeneracy of the states:

n1 g1 = e−E10/kBTS = 3e−E10/kBTS . (1.37) n0 g0 1 The excited state is the triplet state and has a three-fold degeneracy: , ( | ↑↑i √2 | ↑↓ 1 + ), , and the lower-energy state is the singlet state: ( ), where i | ↓↑i | ↓↓i √2 | ↑↓i − | ↓↑i arrows are shorthand for the sign of the z-component of the spin angular momentum for

each of the electron and the proton. TS is the spin temperature and is defined via this equation. For our purposes, T E /k , so we have n /n g /g =3 and S ≫ 10 B 1 0 ≈ 1 0

E10 1 e−E10/kBTS . (1.38) − ≈ kBTS Furthermore, the integral along the line of sight of the number density of hydrogen atoms in the hyperfine ground state will simply be the column density of hydrogen atoms 1 multiplied by 1/4, NHI/4. Here, the factor of 4 accounts for the fact that only one out of four hydrogen atoms will be in the singlet state on average. Putting this together, Eq. 1.36 becomes

NHI E10 τν σ01 φ(ν) (1.39) ≈ 4 kBTS where 3c2A σ 10 (1.40) 01 ≡ 8πν2 is the cross section for 21-cm absorption and A = 2.85 10−15 sec−1 is the spontaneous 10 × emission coefficient for the transition and φ(ν) is the 21-cm line profile. As we discussed in

39 1.2 The Shoulders of Giants

1.2.1.4, line profiles depend on several properties of the gas, however, here it is the case § that Doppler broadening due to the expansion of the Universe dominates the line profile, such that

c φ(ν) (1.41) ≈ sH(z)ν where s is the proper distance between two points expanding away from each other. Putting these pieces together, our expression for the optical depth becomes:

2 3c A10 hν c nHI xHI s τν 2 h i (NHI = snHI xHI ) ≈ 8πν kBTS sH(z)ν 4 h i 3 3hc A10 nHI xHI 2 h i (1.42) ≈ 32πν kBTS H(z)

plugging in values and evaluating at line center, Furlanetto et al. (59) obtain

x H(z)/(1 + z) τ 0.0092(1 + δ)(1 + z)3/2 h HIi (1.43) ν0 ≈ T dv /dr S  k k 

where dvk/drk is the gradient in the line-of-sight velocity (peculiar velocity and velocity due to Hubble expansion) and the ratio of that with H(z)/(1 + z) represents the deviation from pure-Hubble expansion. For the purposes of the 21-cm probes we will discuss, we actually care about the contrast between the 21-cm signal and the background CMB signal. Thus, the relevant brightness temperature contrast, in our frame, is

40 1.2 The Shoulders of Giants

1 −τ −τ δT = T (1 e ν0 )+ T e ν0 T (1.44) b 1+ z S − CMB − CMB TS TCMB −τ  = − (1 e ν0 ) (1.45) 1+ z − T T S − CMB τ (1.46) ≈ 1+ z ν0 T H(z)/(1 + z) 9mK x (1 + δ)(1 + z)1/2 1 CMB (1.47) ≈ · HI − T dv /dr  S  k k  1+ z 1/2 T H(z)/(1 + z) 24mK x (1 + δ) 1 CMB . (1.48) ≈ · HI 7 − T dv /dr    S  k k  This shows that for a neutral parcel of hydrogen at mean density and z = 6 and with T T , the brightness temperature contrast is 24mK. We can also see, however, that S ≫ CMB ∼ this expression depends on several astrophysical quantities other than the neutral fraction. During the course of reionization, the changes in the signal are driven by changes in the neutral fraction, however, in 1.2.2.4, we discuss the redshift ranges when different terms become important for fluctuations in the 21-cm signal.

41 1.2 The Shoulders of Giants

Figure 1.11: Schematic representation of the 21-cm transition where the transition between aligned spins of the proton and electron to anti-aligned spins results in the emission of a photon with λ = 21cm.

42 1.2 The Shoulders of Giants

Figure 1.12: Simulation cube of the 21-cm signal during reionization (top-left) along with simulated noise for an interferometer (top-right) and the galactic foregrounds (bottom). This figure demonstrates that, while the sources of noise are several orders of magnitude larger than the signal, these three contributions to observations are dominant on different scales. The volume of each cube is 1 (Gpc/h)3. In this figure, the line of sight direction away from the observer is to the right and slightly out of the page.

43 1.2 The Shoulders of Giants

1.2.2.2 21-cm Fluctuations with Interferometers

The first approach for utilizing the 21-cm line in order to learn about reionization that we will discuss is through the use of 21-cm fluctuations. The ultimate goal for the 21-cm line regarding reionization is to image the 21-cm intensity field throughout its duration. Since the 21-cm radiation of interest is emitted at z 10, by the time it arrives to us, its ≈ wavelength will have redshifted to 2m. Thus, observations of 21-cm fluctuations must be ∼ made with large interferometers, consisting of tens to thousands of antennae, and maximum separations (baselines) of the order 1 km. Before continuing, it may be helpful to provide a brief description of what interferometry is, for which we will follow Morales and Wyithe (128). Interferometers are built of many an- tennae which observe the electric field at their location and correlate those measurements to image the sky. Let us consider a single antenna which observes a time-and-angle-dependent electric field, defined E(θ,t). A nearby antenna, separated from the first by the vector ~r, will observe a similar electric field, but the field will have a phase shift owing to the difference in path length that the radiation had to travel between going to the first and the second antennae. If we are observing directly overhead, then light will have to travel an equal distance to reach either antenna. Conversely, if we are observing at the horizon along the line connecting the antennae, then there will be a time difference, ∆t = r/c, between when the light reaches the two antennae. This is illustrated in Figure 1.13. Based on this figure, we can see that the extra path length for the light will be

∆ℓ = ~r θˆ (1.49) · resulting in a phase difference of

~r θˆ ∆φ = 2π · . (1.50) λ

44 1.2 The Shoulders of Giants

Therefore, if the first antenna sees an electric field E(θ,t), then the second antenna will −2πi(~r·θ/λˆ ) 1 see E2(θ,t) = E(θ,t)e . The observed electric field at any position, r, is then related to that at the first antenna by

ˆ E(~r, θ, t)= E(θ,t)e−2πi(~r·θ/λ). (1.51)

If we integrate over the entire sky, we find

ˆ E(~r, t)= d2θE(θ,t)e−2πi(~r·θ/λ). (1.52) Z The expression on the right is simply a Fourier transform. Therefore, we see that the observed electric field on the surface of the is actually just a Fourier transform of the electric field on the surface of the sky. This is basically what allows measurements at many locations on the ground to be converted into measurements of the 21-cm signal across the sky. Furthermore, each pair of antenna essentially measure a particular Fourier mode, k, on the sky:

2πu k = (1.53) D where u denotes the separation of a pair of antenna in units of the observed wavelength, D is the co-moving distance to the source of the emission, and k is akin to a spatial frequency at the location of the emission. Thus, this demonstrates that many measurements on the ground can give us information of the 21-cm signal as a function of position on the sky, with widely-separated antenna providing information on small scales and closely-separated an- tenna providing information on the large scales. Several experiments are currently planned

1Throughout this work, we assume we are observing over a sufficiently-narrow field of view such that we may neglect the curvature of the sky.

45 1.2 The Shoulders of Giants

or underway with the goal of measuring this signal. These include the Precision Array for Probing the Epoch of Reionization (PAPER, Parsons et al. 137, underway), the Murchison Widefield Array (MWA, Tingay et al. 174, underway), the Hydrogen Epoch of Reioniza- tion Array (HERA1, planned), the Square Kilometer Array (SKA2, scheduled construction 2018), the Low Frequency Array (LOFAR, Yatawatta et al. 188, underway), and the Giant Metrewave Radio Telescope (GMRT, Paciga et al. 134, underway). These experiments are further discussed in 1.2.2.3. § However, before these experiments can uncover the cosmological 21-cm signal, they must first overcome several formidable challenges. We will only discuss a couple of these challenges here, but refer the reader to, for example, (128) for a more comprehensive dis- cussion. The first challenge is to remove contamination from various sources of foreground emission, which are many orders of magnitude brighter than the redshifted 21-cm signal of interest. Synchrotron emission, free-free emission, and Bremsstrahlung radiation from our own galaxy constitute the largest nuisance at the frequencies of interest (ν 140 MHz) in ≈ sheer magnitude. In a relatively “cold” spot on the sky, Furlanetto et al. (59) state that the intensity of this radiation is a power-law in frequency of the form:

ν −2.6 T 180 K, (1.54) sky ∼ 180 MHz   which scales to T 130 K at z = 6. Thus, we see in this specific example, the sky ∼ intensity of the foreground emission from our galaxy should be & 5000 times greater than the 21-cm signal itself. Furthermore, while it is often the case that sources of noise can be beaten down with increased observation time, these galactic foregrounds are a permanent feature of the sky. While this presents an enormous challenge for observing the 21-cm signal from the EoR, not all hope is lost.

1http://reionization.org/ 2https://www.skatelescope.org/

46 1.2 The Shoulders of Giants

We can see from Eq. 1.54 that the foreground noise from the galaxy should vary very smoothly with frequency. Meanwhile, for the 21-cm signal, slight decreases in observed fre- quency are equivalent to observing the 21-cm signal from a more distant point in space. For example, in the case of ionized bubbles of size ℓ 10 Mpc/h, we would expect fluctuations ≈ of 100% in the 21-cm signal over this distance. However, this distance at a redshift of z 7 ≈ corresponds to a frequency change of ∆ν 1 MHz and a change in the amplitude of the ≈ foreground signal of only . 3%. This is further demonstrated by Figure 1.12. The top-left cube shows what the 21-cm signal would look (in arbitrary units) like for a 1Gpc/h cube of the Universe from a reionization simulation. We can see bubbles of size 10 Mpc/h and ∼ therefore fluctuations in the 21-cm signal on that scale. In the bottom panel, we show a simulation of the galactic foregrounds (provided by Piyanat Kittiwisit at ASU) correspond- ing to the same physical region in space. In this case, the dimension of the cube that goes to the right and slightly out of the page is the line of sight direction away from us. We see that, along this direction, the foreground emission varies very smoothly, as expected from Eq. 1.54. Thus, in principle, the contributions to the observed signal from the EoR and from the galaxy should separable. This is essentially done by excising modes with a smaller wavenumber in the line-of-sight/frequency direction, i.e. by removing spectrally-smooth components. This should be effective at removing the foregrounds but comes at the price of removing the underlying 21-cm signal at those wavenumbers as well. See 5.2.4 for a brief § discussion. A second challenge relates to achieving the sensitivity required to measure the faint spatial fluctuations in the redshifted 21-cm emission from the EoR. This is discussed in more detail in 5.2.3, but the power spectrum for thermal noise in the interferometer should § follow (McQuinn et al. 113,Furlanetto et al. 59)

T 2 D2∆D λ2 2 P = sky . (1.55) N B t n(k ) A int ⊥  e 

47 1.2 The Shoulders of Giants

Here, k⊥ is the component of k transverse to the line of sight, B is the bandwidth of the observation, tint is the integration time for the experiment, n(k⊥) is the number density of baselines observing the specific wavemode, ∆D is the depth of the observation

field, and Ae is the effective collecting area per tile. With the exception of Tsky and λ and D, these are controllable parameters of the experiment. This allows for several avenues toward beating down the thermal noise, the most popular of which are through increasing observation time, re-arranging tiles in order to alter n(k⊥), and increasing collecting area. Thus, unlike the galactic foregrounds, the thermal noise is a nuisance which will become less and less important as experiments evolve. The expected level of thermal noise in first-generation experiments should mostly pro- hibit constructing detailed maps of the redshifted 21-cm signal. This is discussed in Mc- Quinn et al. (113) and shown in Figure 1.14. Specifically, this figure shows the fraction of pixels at each k which will be imaged (have SNR 1) for MWA (dashed), LOFAR (dot- ≥ dashed), and SKA (solid) assuming galactic foregrounds can be subtracted. With regard to foregrounds, though, it is likely that any k modes below the vertical hatched line will be inaccessible due to residuals from the foreground subtraction. This demonstrates that, due to thermal noise, first-generation interferometers like the MWA and LOFAR will image a small fraction of the pixels at the k modes unspoiled by foregrounds and will therefore be unable to make maps of the 21-cm sky during reionization.1 In Figure 1.12 we show a random, qualitatively-representative simulated realization of thermal noise in the top right. While this panel indeed does demonstrate that thermal noise should dominate the 21-cm signal (top-left) in sheer magnitude, it also demonstrates that the fluctuations in the thermal noise are expected to happen on scales smaller than those

1This estimate is done assuming that fluctuations in the 21-cm signal are caused by density fluctuations rather than by fluctuations in the ionization field. The latter should lead to a larger signal and so, in this sense, this estimate is somewhat pessimistic. On the other hand, this figure is generated assuming out-of- date experiment configurations, which have since been downgraded in some cases. In any case, we make improved forecasts of the imaging capabilities in §5.

48 1.2 The Shoulders of Giants

in the 21-cm signal. While galactic foregrounds ruin large-scale modes, and thermal noise ruin small-scale modes, an intermediate region of k-space should remain accessible to the interferometers. It will not be the case that first-generation interferometers can image or make movies of the 21-cm signal, but they still may learn about how the signal behaves on different physical scales, i.e. the power spectrum, and how that behavior evolves with redshift. Such an approach can take advantage of the fact that, while the signal to noise in each Fourier mode is generally expected to be small, one may still be able to detect the power spectrum by binning together many individually-noisy modes. In fact, Lidz et al. (90) investigate the evolution of the power spectrum throughout reionization under several simulated reionization scenarios. They find that a generic result to all the models is that the slope of the power spectrum, in the accessible k-mode range (0.1 h/Mpc k 1 h/Mpc) should decrease as reionization evolves. Meanwhile, the ≤ ≤ amplitude of the power spectrum in this k-mode range should rise until reionization reaches its midpoint (defined here as the redshift at which 50% of the volume is in the ionized ∼ phase) and then fall. Thus, measuring the 21-cm power spectrum at several redshifts and confirming this behavior can, first, increase our confidence that we are in fact observing the 21-cm signal from reionization and, second, help us learn about the reionization process itself. Constraints have already been placed on the nature of the EoR using this approach by, for example, the PAPER collaboration. In Ali et al. (3) and Pober et al. (145), upper limits on the amplitude of the 21-cm power spectrum at z = 8.4 are converted to constraints on the IGM properties. Specifically, we see in Eq. 1.48 that, if T T , then the amplitude of S ≪ CMB the 21-cm signal can become arbitrarily large. As we will touch on in 1.2.2.4, for redshifts § relevant to reionization, the spin temperature is tied to the gas temperature. As such, upper limits on the 21-cm power spectrum amplitude can be converted to lower limits on the spin temperature and therefore also lower limits on the gas temperature. These authors use the upper limits to rule out a very cold reionization scenario.

49 1.2 The Shoulders of Giants

Measurements of the 21-cm power spectrum and its evolution are powerful probes of the Epoch of Reionization, but they are inherently indirect. Making detailed maps of the 21-cm radiation field would be much more direct, but will likely be out of reach for first and second generation experiments. However, the presence of an intermediate range in k space which remains relatively un-spoiled by noise could provide us with the ability to make crude maps of the 21-cm field and/or directly image individual large ionized regions. Direct observations of individual ionized regions would provide unambiguous evidence that reionization is ongoing and could also provide hints as to the sources of the ionizing photons and the volume-averaged neutral fraction. Such proposed approaches generally attempt to reconstruct the underlying signal by downweighting the k modes of the observed signal which are expected to be dominated by noise. While these techniques are out of reach for first-generation experiments like PAPER, they may very well be suitable for successor experiments, such as HERA. We develop some of these filtering approaches in 5 and explore § their utility for a variety of plausible experiments.

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Figure 1.13: Depiction of the extra path length, ∆ℓ, of radiation (dot-dashed lines) incident on two elements (solid black rectangles) in an array separated by ~r when considering a position on the sky θˆ.

51 1.2 The Shoulders of Giants

1

0.8

0.6

0.4

0.2 Fraction of Imaged Pixels at k 0 −2 −1 0 10 10 10 k (Mpc−1)

Figure 1.14: Percentage of pixels “imaged” (SNR > 1) as a function of wavemode, k for the MWA (dashed), LOFAR (dot-dashed), and the SKA (solid). The vertical hatched line shows the distance scale above which (smaller k) the residuals from foreground subtraction are expected to dominate the 21-cm signal. This demonstrates that, for first-generation 21-cm experiments, a very small fraction of pixels with k > khatched will be “imaged”. This estimate assumes that fluctuations in the 21-cm signal are driven from density fluctuations rather than fluctuations in the ionization field, so it is somewhat conservative. Taken from McQuinn et al. (113).

52 1.2 The Shoulders of Giants

3 Fiducial Massive Mini-Halos 2 Density

1

0

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1

Figure 1.15: The redshift evolution of the 21-cm power spectrum in simulated models of reionization. The left panel shows the evolution of the power spectrum during reionization for the fiducial reionization model in Lidz et al. (90). We can see that, as reionization progresses, the slope of the power spectrum in the k-mode range accessible to interferometers (0.1 h/Mpc k ≤ ≤ 1 h/Mpc) declines. The amplitude of this part of the power spectrum peaks around xHI =0.5. h i The right-hand panel shows the evolution of the power spectrum slope (top) and magnitude (bottom) during reionization for a few different reionization models. This demonstrates that the general power-spectrum evolution described is generic to many reionization models. Both figures are taken from Lidz et al. (90).

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1.2.2.3 Brief Description of 21-cm Interferometric Experiments

In this section, we provide a brief description of the aforementioned interferometers aimed at detecting the 21-cm signal from the Epoch of Reionization. Much of the descriptions will closely follow information on their respective websites, which are also provided. The Giant Metrewave Radio Telescope (GMRT1) is a collection of 30 very large steerable antennae with a diameter of 45m and is located 80km north of Pune, India. Approximately half of the antennae are configured in a dense central core, with the remaining antennae forming a very extended “Y” shape, with the largest baseline being 25km. This multi- purpose experiment has been operating since 1995 and is interested in investigating 21-cm emission from the formation of the first galaxies and reionization, learning about pulsars and neutron stars, and studying properties of the , among many other topics. As such, the configuration is not optimized for studying the 21-cm emission from the EoR. A picture of a few of the antennae are shown in Figure 1.16.

Figure 1.16: Several antennae in the GMRT core. www.mso.anu.edu.au

The Donald C. Backer Precision Array to Probe the Epoch of Reionization (PAPER) is a radio interferometer built to detect the power spectrum from 21-cm emission during the EoR. The primary science array is located in the Karoo desert of the Northern Cape in

1http://gmrt.ncra.tifr.res.in/

54 1.2 The Shoulders of Giants

South Africa. The experiment initially deployed 16 antennae in 2009 with the intention of increasing the number of array elements with time. Their most recent constraints (Pober et al. 145) on the EoR were made using a 64-element configuration, which is currently being expanded to 128 elements. Some of the elements of the array are shown in Figure 1.17, which demonstrates one of the the highly-redundant configurations, with many baselines observing the same k-mode, optimized for power-spectra measurements.

Figure 1.17: A highly-redundant configuration of tiles for the PAPER interferometer, well- suited for power-spectrum measurements. Picture from www.discovermagazine.com

The Murchison Widefield Array (MWA1) is a radio interferometer located in the Shire of Murchison in Western Australia. This is a multi-purpose experiment aimed at investigating the EoR, galactic science, transient sources, and space weather. It is composed of 128 antenna tiles, which are each composed of 16 dipole antennae. The majority (112) of the tiles are located within a 1.5km region, with the remaining tiles at larger separations in

1http://www.mwatelescope.org/

55 1.2 The Shoulders of Giants

order to facilitate calibration and to pursue other science goals. The MWA began science observations in 2013. An image of some of the antenna tiles which compose the central core is shown in Figure 1.18. The array configuration shows much less redundancy compared with PAPER, with tiles placed seemingly-randomly in a configuration more favorable for imaging. The MWA, together with PAPER, is a pathfinder for HERA (described below) and is also referred to as HERA Phase I.

Figure 1.18: Several antenna elements in the core of the MWA array. Image taken from www.mwatelescope.org/multimedia.

The Hydrogen Epoch of Reionization Array (HERA1 is an array in preparation which will use understanding gained from both PAPER and MWA in order to make statistical detections and images of the Epoch of Reionization. A potential layout of the HERA experiment is shown in Figure 1.19, displaying a hexagonal arrangement of 331 elements, each with a 14-meter diameter. As such, this represents an order of magnitude increase in collecting area compared to first-generation experiments. The Low-Frequency Array (LOFAR2) is another currently-operational multi-purpose interferometer taking strides toward making a statistical detection of the EoR. In addition 1www.reionization.org 2www.lofar.org

56 1.2 The Shoulders of Giants

to the EoR, the array is also aimed at making deep extragalactic surveys, studying transit radio phenomena, high energy cosmic rays, cosmic magnetism and space weather.1 The core of the array is located in the Netherlands but stations of the array are also located in Germany, Great Britain, France, and Sweden. The instrument began observations in late 2012. An image of the central antenna stations is shown in Figure 1.20. Compared to the other arrays discussed thus far, we see LOFAR has larger individual tiles, in a non- redundant configuration, with a significantly larger minimum separation. As such, it will be less sensitive to small k-modes, according to Eq. 1.53. Together with the MWA, LOFAR is a pathfinder experiment for the Square Kilometer Array. Lastly, the Square Kilometer Array (SKA2) is another later-generation interferometer. Construction is currently expected to begin in 2018, with a planned first light of 2020. The antenna stations will be located in South Africa and Australia. It is another multi- purpose experiment. The reionization component will make observations of the frequency interval 50MHz ν 350MHz, corresponding to 3 z . 30, in principal allowing for ≤ ≤ ≤ observations prior to the Epoch of Reionization. An artist’s impression of what the low- frequency reionization-focused component of the experiment might look like is shown in Figure 1.21.

1http://en.wikipedia.org/wiki/LOFAR# Key projects 2https://www.skatelescope.org/

57 1.2 The Shoulders of Giants

Figure 1.19: Planned layout of the HERA interferometer. Image taken from (45).

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Figure 1.20: The central antenna stations for the LOFAR interferometer. Image taken from www.astron.nl.

Figure 1.21: An artists impression of what the reionization-focused element of the SKA might look like. “SKA sparse array big” by SKA Project Development Office and Swinburne Astronomy Productions - Swinburne Astronomy Productions for SKA Project Development Office. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

59 1.2 The Shoulders of Giants

1.2.2.4 The Global 21-cm Signal

An alternative method of extracting information from the 21-cm line is to use the sky- averaged 21-cm signal. In the previous section, we discussed the difficulties faced in obtain- ing the necessary resolution to map out fluctuations in the 21-cm radiation field. However, Equation 1.48 is rich with astrophysical information on its own without even considering spatial fluctuations. Therefore, natural questions to ask would be if it is easier to simply measure the average signal rather than map the fluctuations and what astrophysical infor- mation can be obtained in that way? For convenience, the brightness temperature contrast for the 21-cm signal from 1.2.2.1 is: §

1+ z 1/2 T H(z)/(1 + z) δT 24mK x (1 + δ) 1 CMB . (1.56) b ≈ · HI 7 − T dv /dr    S  k k  As we discuss subsequently, the current expectation is that X-ray heating will boost the kinetic temperature of the gas in the IGM to much above the CMB temperature significantly before reionization completes. This may happen throughout much of the IGM volume by the time only 10% of the volume is ionized. At this point, the Wouthuysen-Field effect ∼ (see below) should also have succeeded in coupling the gas kinetic and spin temperatures, and so the spin temperature will be globally larger than the CMB temperature. In this case, the spin temperature factor will (approximately) drop out of Eq. 1.48 ( 2 of Furlanetto et al. § 60, and references therein). The result is that the amplitude of a sky-averaged signal is most sensitive to the neutral faction, with ionized regions emitting no 21-cm signal and neutral regions emitting 21-cm radiation with brightness temperature of tens of mK. Thus, one could imagine plotting the sky-averaged 21-cm signal throughout the EoR against redshift and observing a shrinking signal toward lower redshifts coinciding with a larger fraction of the volume of the Universe being ionized. However, the utility of the global 21-cm signal is not limited to the Epoch of Reioniza- tion. Specifically, prior to reionization, x will be fixed at 1 and the spin temperature h HIi

60 1.2 The Shoulders of Giants

is expected to drop to the point where 1 T /T 1. Therefore, the strength of the − CMB S 6≈ 21-cm radiation field will be tied to the spin temperature instead of the neutral fraction. Furthermore, the 21-cm signal will only be observable if T = T and will appear in S 6 CMB emission if TS > TCMB and will appear in absorption otherwise. The hydrogen spin temper- ature is determined by several competing processes, such as absorption of CMB photons, collisions with other hydrogen atoms, electrons, or protons, and scattering of UV photons. Specifically, the spin temperature in equilibrium will satisfy (Field 55, Furlanetto et al. 60)

−1 −1 −1 −1 TCMB + xcTK + xαTc TS = , (1.57) 1+ xc + xα

where xc is the collisional coupling coefficient, xα is the UV scattering coupling coef-

ficient, and Tc is the “color” temperature and is related to the UV radiation field. This is interesting because the coupling coefficients determine the spin temperature and are themselves dependent on several astrophysical processes. A very high-level description of the evolution of the 21-cm signal due to these astrophysical processes is shown in Figure 1.22.1 The precise evolution of the spin temperature is not well known, but a reasonable approximate description could be as follows:

z > 150: Collisions within the gas are frequent enough to fix the spin temperature to

the gas temperature, TS = TK. However, the gas temperature is itself coupled to the CMB temperature, so the 21-cm signal is unobservable.

30

1This figure and much of the following discussion is taken from notes from Adam Lidz’s “Topics in Cosmology” class, taught in the Fall of 2011.

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20

15

z < 15 : The first X-ray sources form and heat the gas well above the CMB tempera- ture. As a result, the 21-cm signal becomes observable in emission.

z . 5.5 : The completion of reionization effectively eliminates the 21-cm signal from the IGM all together. The 21-cm signal is still observable from galaxies and DLAs which retain self-shielded neutral gas after reionization.

While the expected globally-averaged 21-cm signal during z > 30 (and z < 5) is well- understood, much of the rest of the signal is very uncertain. Therefore, we could imagine measuring δT¯b(z) in order to constrain some of the underlying astrophysics during this time period. However, as we discussed in 1.2.2.2, the foreground noise is several orders of § magnitude larger than the 21-cm signal during the EoR and varies smoothly with frequency. When considering 21cm fluctuations, we were saved by the fact that the signal fluctuated rapidly along any individual line of sight. However, when considering the average 21-cm signal, fluctuations should be very smooth along the line of sight. Specifically, from the end of reionization to the beginning, the global 21-cm signal should increase from δT¯b = 0 to δT¯b & 30mK. As such, a drawn-out and extended reionization scenario will be relatively difficult to detect. However, in the case of a rapid reionization

62 1.2 The Shoulders of Giants

Figure 1.22: Schematic representation of the 21-cm signal. The top panel shows a plausible signal for 21-cm fluctuations from shortly after the big bang (left) to today (right). Blue indicates the signal is seen in absorption and red indicates it is seen in emission. In the bottom panel, the strength and sign of the averaged signal is shown along with several important landmarks coinciding with the turning points in this curve. The redshift is shown at the top of the bottom panel. The precise timing of the turning points is not well-constrained, this is just one plausible history. As such, the exact redshift values do not completely match those that we described in the text. Figure taken from (147).

63 1.2 The Shoulders of Giants

of the Universe, it is conceivable that this evolution could be quite steep. Bowman and Rogers (26) measured the global 21-cm signal for three months in the frequency range 100MHz ν 200MHz, corresponding to 6 0.06 with 95% confidence. Another experiment which has gained some attention in this sector is the (DARE, Burns et al. 27) which aims to measure the turning points and slope of the 21-cm signal (shown in Figure 1.22) over the redshift range 11 z 35 in ≤ ≤ order to constrain the formation of the first stars, galaxies, accreting black holes along with the amount subsequent X-ray heating, and constrain the beginning of the EoR.1 This will be done by placing a 21-cm antenna in lunar orbit. It is interesting to note that, even at this seemingly-ideal observation location, much effort will be needed to isolate nuisance contributions to the 21-cm signal, such as those from foreground emission from our own galaxy, the Sun, thermal emission from the Moon, and reflected galactic foregrounds off of the surface of the Moon (Harker et al. 69).

1This information about DARE was obtained from http://lunar.colorado.edu/

64 1.2 The Shoulders of Giants

1.2.3 The Cosmic Microwave Background

The Cosmic Microwave Background (CMB) is the earliest accessible picture we have of the Universe. It is composed of photons which have travelled, largely uninterrupted, from when the Universe was 380, 000 years old to us today. Before this time, the Universe ∼ was hot enough that if atoms momentarily formed, they were immediately stripped apart, mostly from collisions with photons above the ionization threshold. As a result, photons had an extremely short mean free path between collisions with free electrons. This makes this period of time completely opaque and impenetrable by observations. However, at 380, 000 years, the Universe cooled to the point where neutral hydrogen could form ∼ (neutral helium having formed slightly earlier) and capture almost all of the remaining free electrons. Without a substantial number of free electrons, the mean free path of photons increased to the point where they could reach us today. Because of this, the CMB photons propagate to us from a so-called “surface of last scattering”. Therefore, the CMB is essentially a picture of the Universe at the time when photons decoupled from matter. As a result, it provides us with a picture of the matter distribution at this time and its value to cosmology cannot be overstated. While this is wonderful, in this thesis we are interested in the evolution of the Universe when it was 500 million ∼ years old, not 380, 000, so how can the CMB help us? ∼ Well, similarly to how bright light from distant quasars/GRBs can give us information on the intervening gas, light from the CMB can do the same. While & 90% of photons will travel from the surface of last scattering to us without interacting, the other . 10% of the photons will scatter off of free electrons that have been released as a result of reionization. Therefore, an interesting question is if this level of interaction can create an observable imprint in the CMB itself. In this section, we briefly discuss two such methods for searching for these imprints and their progress toward constraining the EoR.

65 1.2 The Shoulders of Giants

1.2.3.1 Thomson Scattering Optical Depth, τe

The first observable we discuss is the optical depth of CMB photons to Thomson scattering off of free electrons produced during and after reionization. As mentioned earlier, just after leaving the last-scattering surface, CMB photons encounter a low enough density of free electrons to propagate unimpeded. However, once reionization is underway, free electrons will be introduced into the IGM and will scatter a significant number of the CMB photons. The precise fraction of CMB photons that scatter in this way depends on the integrated electron density along the line of sight to the CMB. This is sensitive to the timing of reionization since CMB photons will have the opportunity to scatter off of electrons over a larger path length if reionization happens earlier. The percentage of CMB photons which

scatter in this way is quantified through the Thomson scattering optical depth, τe, where

f = 1 e−τe τ (1.58) scattered − ≈ e

1 for small τe. This is related to the density of electrons along the line of sight via

dz c τ = n (z)σ (1.59) e (1 + z)H(z) e Thom Z dz c n¯ (z)(1 x (z) )σ (1.60) ≈ (1 + z)H(z) H − h HI i Thom Z where

8π α~ 2 σ = 6.65 10−25cm2 (1.61) Thom 3 m c ≈ ×  e  is the frequency-independent cross section for Thomson scattering, α = 1/137 is the fine

structure constant, ~ is the reduced Planck’s constant, and me is the mass of the electron.

1 We have neglected the contribution to ne from singly-ionized helium in this expression for simplicity.

66 1.2 The Shoulders of Giants

The integral is carried out from z = 0 to the surface of last scattering (z 1080). Thus, we ≈ see that a measurement of the optical depth of CMB photons to Thomson scattering off of electrons released during reionization provides and integral constraint on x . Typically, h HIi such measurements of τe are converted into a redshift of “instantaneous reionization”, zr, by performing the integral assuming x = 0 for z < z and x = 1 for z > z and h HIi r h HIi r finding which value of zr produces the same τe. The instantaneous model is not to be taken seriously, but nevertheless provides a rough indication of the typical redshifts at which reionization occurs.

In order to measure τe we can take advantage of the tendency for photons to have their polarization altered when they undergo Thomson scattering. Specifically, photons propagate such that their E-field and B-field oscillate in directions perpendicular to their direction of travel. When a photon gets scattered into our line of sight, it maintains its original polarization in the plane perpendicular to the line of sight. As such, the net polarization from a point on the sky depends on the in the radiation field present in that point of the sky. This is illustrated in a diagram by Wayne Hu in Figure 1.23. This shows a quadrupole variation in the intensity of radiation incident on a free electron. In this case, the radiation intensity incident from the left/right is greater than that from above/below and this generates a net vertical polarization after Thomson scattering. The electrons sourcing the polarization signal will see radiation from the quadrupole isotropy on scales of the horizon size at the time of the scattering. Therefore, fluctuations in the observed polarization signal will occur on spatial scales equal to the horizon size at the time of reionization. This allows measurements of the polarization power spectrum to be

translated into a measurement of τe. Current measurements from Collaboration et al. (36) determine τ = 0.066 0.0121, which translates to z = 8.8+1.2. Interestingly enough, this is e ± r −1.1 a substantial shift from the first-year WMAP measurement of τ = 0.166+0.076, z = 17 4, e −0.071 r ± and even a substantial shift from the nine-year WMAP results of τ = 0.089 0.014, e ± z = 10.6 1.1 ( Bennett et al. 15). This contributes to some of the accumulating evidence r ±

67 1.2 The Shoulders of Giants

that reionization may have occurred later (lower z) than was originally believed and makes measurement techniques applicable to z . 6 more interesting.

68 1.2 The Shoulders of Giants

Figure 1.23: An illustration (Wayne Hu, http://background.uchicago.edu/ whu/)ofhowa net polarization signal is generated from Thomson scattering due to the presence of a quadrupole anisotropy. The blue cross and red cross show relatively strong and weak incident radiation, respectively, on an electron at the origin. The red/blue cross indicates the average polarization of scattered light and demonstrates that it obtains a net vertical polarization.

69 1.2 The Shoulders of Giants

1.2.3.2 Kinetic Sunyaev-Zel’dovich Effect

The second method of utilizing the CMB to constrain reionization that we discuss is the kinetic Sunyaev Zel’dovich effect (kSZ). This refers to the secondary CMB anisotropies imprinted on the CMB by the bulk velocities of clouds of free electrons which impart a Doppler shift on the CMB photons. The kSZ signal is generally broken down into two contributions: the Doppler shift caused by bulk motions of free electrons after reionization completes, and the Doppler shift due to the bulk motions of free electrons in ionized bubbles during reionization. As such, the former contribution is known as the “homogeneous” (or, alternatively, Ostricker-Vishniac [OV]) kSZ signal and the latter is referred to the “patchy” kSZ signal. The homogeneous kSZ signal is sourced by density inhomogeneities on relatively linear scales and should be able to be modelled well (Mesinger et al. 121), allowing the isolation of the patchy contribution. The patchy kSZ signal is actually sensitive to several parameters of reionization. First, unlike with τe measurements, the patchy kSZ contribution only arises during reionization. Therefore, the magnitude of the signal itself is related to the duration of reionization since, for a longer reionization, CMB photons will have an opportunity to be Doppler shifted by ionized bubbles over a larger path length. Second, since the patchy kSZ signal is sourced by ionized bubbles, it is sensitive to the size of the ionized regions and the strength of the signal will also increase with patchier models of reionization. As such, if the kSZ signal is found to be smaller than expected, this might indicate a more homogeneous reionization process; this could, for example, imply a significant contribution of ionizing photons from X-ray emitting sources (Visbal and Loeb 179). The South Pole Telescope (SPT) attempted a measurement of this effect (Zahn et al. 191) and interpreted the results in terms of a constraint on the duration of reionization, ∆z z z . They were not able to make a detection of the effect but were ≡ hxHIi=1 − hxHIi=0.2 able to place upper bounds on it, suggesting that ∆z < 7.9 at 95% confidence. Technically, this constraint allows for a free parameter describing the level of correlation between the

70 1.2 The Shoulders of Giants

thermal SZ effect and the Cosmic Infrared Background. Without allowing for this free parameter, their constraint is ∆z < 4.4 at 95% confidence.

71 1.2 The Shoulders of Giants

1.2.4 Ly α Emitters

Ly α emitters (LAEs) are galaxies which emit strongly in the Ly α line. This Ly α emission results from hydrogen atoms within the galaxy recombining after being ionized by the galaxy’s UV radiation. During 2/3 of hydrogen recombinations, a Ly α photon will be ∼ emitted. Therefore, enough ionizations will result in a strong Ly α line being emitted from the galaxy. However, whether or not that Ly α line is observable to us depends on the intervening gas. Specifically, if the IGM surrounding an LAE is highly-ionized, then photons at the Ly α frequency and redward should escape and travel unimpeded through the IGM. However, if the IGM surrounding the LAE is neutral, the corresponding damping-wing absorption will wipe out the Ly α line entirely. This provides us with an observable which depends on the ionization state of the IGM! In this section, we briefly discuss two methods of utilizing this behavior in order to constrain x . h HIi

1.2.4.1 Clustering of Ly α Emitters

One approach for utilizing LAEs to constrain x is to look at their measured clustering. h HIi When the IGM is fully-ionized, the Ly α lines of all observed galaxies should be visible to us. When the Universe is fully-neutral, then all observed galaxies should lack a strong Ly α emission line. However, when reionization has progressed such that x 0.5, the h HIi ≈ Universe should represent a two-phase medium consisting of large ( 10 Mpc/h) ionized ∼ regions maintained by thousands of galaxies (McQuinn et al. 108), and significantly-neutral regions which are shielded from the ionizing radiation. The Ly α line from LAEs located within the large ionized regions should remain intact, as the photons will redshift out of Ly α resonance after travelling only 1 Mpc/h without encountering significantly-neutral ∼ hydrogen (Finlator 56). Therefore, the LAEs observable when the Universe is 50% ionized should more often reside in these ionized bubbles with many other sources, resulting in their

72 1.2 The Shoulders of Giants

observed distribution being highly clustered. Meanwhile, at lower redshift, all LAEs should be observable (with regard to HI attenuation), resulting in a more uniform distribution. In Figure 1.24, McQuinn et al. (108) demonstrate this effect. The top row shows the ionization field (black is neutral, white is ionized) at three different neutral fractions: x = 0.7 (left), 0.5 (middle), and 0.3 (right). The second row shows the true underlying h HIi LAE locations and the bottom row shows the observed LAEs. Since LAEs should only be observable within ionized regions, we see that the sources in the bottom row must coincide with white regions in the top row, resulting in enhanced clustering of the observed sources. In Ouchi et al. (131), the authors analyze 207 LAEs at z 6 7 and compare their ∼ − clustering with those measured at z = 5.7. They find no detection of an enhancement in the observed LAE clustering, suggesting that the bulk of reionization occurred at z > 6.6.

73 1.2 The Shoulders of Giants

x i= .3 x i=.5 x i= .7

Intrinsic

Observed

Figure 1.24: The (simulated) effect of the neutral fraction on the observed clustering of LAEs (taken from McQuinn et al. 108). The top panels show the underlying ionization fields, the middle row shows the true location of LAEs in the simulation, and the bottom panel shows the detectable LAEs in the simulation. This shows that, LAEs which occupy the same ionized bubble will be observable, resulting in a less homogeneous field of observable LAEs. Each panel is 94 Mpc across.

74 1.2 The Shoulders of Giants

1.2.4.2 Ly α Emitter Fraction

As discussed in the previous section, as we move further back in redshift and deep into the reionization process, galaxies that intrinsically have a significant Ly α line will not be observed as having one. However, the galaxies themselves will still be detectable via the drop-out technique, which searches for sources with significant emission at energies below Ly α and significantly less emission at greater energies. As we move further back in redshift, we expect the fraction of detected galaxies which would have a Ly α line but do not, due to a significantly-neutral IGM, will increase. While we do not have access to this exact measurement of this fraction, we can measure the overall fraction of detected galaxies which exhibit a strong Ly α line, which should reflect the aforementioned trend. This has motivated the study of the evolution of the so-called

“Ly α fraction”, denoted fLy α (Caruana et al. 28, 29, Pentericci et al. 141, 142, Schenker et al. 161). This method has the benefit compared to some others, such as measuring the LAE luminosity function evolution, that some of the overall redshift evolution in the intrinsic properties of the observed galaxies, unrelated to the EoR, will drop out. Interestingly enough, some of these authors’ analyses claim to support a surprisingly- neutral IGM. Specifically, work by Caruana et al. (29), Pentericci et al. (142), Pentericci et al. (141), and Schenker et al. (161) all suggest a neutral fraction of x (z 7) 0.5. h HI ∼ i ∼ This seems to be in tension with other constraints on the reionization process. Namely, in 1.2.3, we discussed constraints on the redshift of “instantaneous reionization”, which § +1.3 can be interpreted as an upper bound on the mid-point of reionization, of zr = 8.8−1.2 (Collaboration et al. (36)). Additionally, analysis by Bolton and Haehnelt (19) suggests that reionization is a rather extended process. Assuming this fLyα constraint is correct, this would allow only ∆z 1 for the second half of reionization to complete. ∼ One plausible way to reconcile these observations is presented by Bolton and Haehnelt (20) who suggest that the rise in the prevalence of dense absorbers at high z, rather than a rise in the neutral fraction of the diffuse IGM, could contribute to a surprisingly-small fLyα

75 1.2 The Shoulders of Giants

without requiring changes in the neutral fraction of tens of per cent over z 6 7. Fur- ∼ − thermore, Taylor and Lidz (170) argue that, due to the expected large-scale inhomogeneity of reionization, LAE surveys which sample relatively small regions of the sky are subject to sample variance which can mitigate the high-neutral-fraction requirements. However, their analysis still suggests that x > 0.05 at 95% confidence. Thus, while the precise amount h HIi of neutral hydrogen required to explain the fLyα observations is controversial, it is exciting that the conclusion that we are observing some phase of reionization is rather robust.

1.2.5 Luminosity Function Measurements

One last method that we will discuss for constraining the Epoch of Reionization is through luminosity function measurements. Luminosity functions describe the abundance of sources, in this case galaxies, as a function of their luminosity (here in the rest-frame part of the , near 1350A).˚ With knowledge of the luminosity function, the ionizing luminosity of galaxies within a given luminosity bin, and of the escape fraction of ionizing photons over a range of redshifts, it should be possible to effectively count the number of ionizing photons that are injected into the IGM at different redshifts. From this, one can determine if enough ionizing photons were produced by a given redshift in order to ionize the Universe and keep it that way. As such, the authors in Robertson et al. (155) and Robertson et al. (154) use mea- surements of the number, luminosity, and spectral properties of galaxies observed by the 2012 Hubble Ultra Deep Field Campaign in order to make estimates of the time-dependent cosmic ionization rate:

n˙ ion = fescξionρUV (1.62)

76 1.2 The Shoulders of Giants

wheren ˙ ion is the number of ionizing photons per unit volume per unit time, fesc is the 1 fraction of ionizing photons that escape their host galaxy, ξion is the number of ionizing photons emitted per time per unit luminosity, and ρUV is the UV luminosity density. With an estimate ofn ˙ ion in hand, one can calculate the ionization history using

d x n˙ x h ii = ion h ii. (1.63) dt nH − trec In this expression, the right hand side incorporates the rate of ionizations on the left

and the rate of recombinations on the right, where trec is the recombination time (discussed in 4.3). Armed with estimates of the ionization history, the authors are then able to use § Eq. 1.60 to find the corresponding value for the optical depth of CMB photons to Thomson scattering. Such analyses come to the conclusion that, if -forming galaxies drive reionization, then there must be a significant population of galaxies below the luminosity detection threshold in order to provide enough ionizing photons to ionize the Universe by z 6, even ∼ assuming fesc = 0.2. Furthermore, they find that, in order to match WMAP constraints on τ , low levels of are required at z & 15 25 (Robertson et al. 155). e − However, recent optical depth measurements from Planck result in a substantially lower

τe which Robertson et al. (154) find reduces the requirement of a significant population of star-forming galaxies at z 10. ≫ In Figure 1.25 (taken from Robertson et al. 154) we see several claimed constraints on x during the Epoch of Reionization (markers), most of which we touch on in 1.2, along h HIi § with best fit curves calculated using luminosity functions. The red shaded curve shows the maximum-likelihood model of the neutral fraction (white) with 1σ errors and is consistent with Planck τe measurements. The analogous curve for Robertson et al. (155) is shown

1It is worth pointing out that this quantity is poorly constrained. Robertson et al. (154) find that their measurements strengthen the hypothesis that galaxies drove reionization, but this conclusion requires a high escape fraction; more than 20% of the ionizing photons must be able to escape from the galaxy.

77 1.2 The Shoulders of Giants

in blue, but is in conflict with the WMAP τe constraints. A model that forces the blue curve to satisfy the WMAP τe constraint is shown in yellow. This figure demonstrates that, under some assumptions, the scenario where galaxies dominate reionization is not in conflict with the constraints on the timing of the EoR to date. As such, these analyses claim to strengthen the case that star-forming galaxies played a dominant role in the reionization of the Universe.

78 1.2 The Shoulders of Giants

Figure 1.25: Several claimed constraints on xHI during the Epoch of Reionization (markers), h i most of which we touch on in this section, along with best fit curves calculated using luminosity functions (Robertson et al. 154). The red shaded curve shows the maximum-likelihood model of the neutral fraction (white) with 1σ errors and is consistent with Planck τe measurements. The analogous curve for Robertson et al. (155) is shown in blue, but is in conflict with the WMAP

τe constraints. A model that forces the blue curve to satisfy the WMAP τe constraint is shown in yellow. This figure demonstrates that, under some assumptions, the scenario where galaxies dominate reionization is not in conflict with the constraints on the timing of the EoR to date.

79 1.3 Moving Forward

1.3 Moving Forward

The previous section has served in part to pay tribute to the extraordinary work that has gone into providing us with our current understanding of the Epoch of Reionization. However, it also demonstrates some of the challenges in interpreting previous measurements and motivates the development of additional approaches for constraining the reionization process. Our discussion of the Ly α forest as a tool for constraining the EoR in 1.2.1 highlights § several of the current drawbacks of the approach. Among these, the largest is arguably the degeneracy between different sources of saturated absorption in high-z quasar spectra. If one could find a way to determine if absorption in the Ly α forest was certainly the result of underlying neutral hydrogen, then it would be easier to interpret observations in terms of the overall neutral fraction of the Universe. We highlighted one example of how this has been done in 1.2.1.3: the hydrogen damping wing redward of Ly α. While a § useful approach, this also suffers from drawbacks of its own, namely the limited number of high-redshift spectra suitable for damping-wing searches, the fact that regions surrounding quasars are not expected to be representative of the IGM as a whole, and challenges with accurately fitting the quasar continuum. This helps motivate two approaches we consider in 2.1 Namely, we identify the hydro- § gen damping wing and absorption due to deuterium as “smoking gun” signals of underlying hydrogen and search for them in typical regions of the IGM. While we will not be able to identify individual absorption features, we will demonstrate that, through strategically stacking regions of absorption in quasar spectra, the features should be observable on av- erage. We show that, if the Universe is & 5% neutral at z 5.5, then damping-wing ∼ absorption from neutral hydrogen and absorption from primordial deuterium should leave an observable imprint in the Ly α and Ly β forest, respectively. Furthermore, the presence

1Based on Malloy and Lidz (98).

80 1.3 Moving Forward

of neutral islands should introduce a bimodality into the size distribution of absorbed re- gions. In 3, we present preliminary results of applying these stacking approaches to the § quasar spectra used in McGreer et al. (104). In the preceding section, we also discussed difficulties faced by current approaches in making measurements of the temperature of the high-redshift IGM. Namely, due to the high levels of absorption in the Ly α forest of quasar spectra at these redshifts, traditional methods of inferring the IGM temperature via line fitting are inapplicable except in the highly-complex proximity zones of quasars or at redshifts well past the end of the EoR. This motivates the development of a temperature-measurement technique which is applicable to typical regions of the high-redshift IGM. In 41 we model the temperature of the intergalactic § medium after reionization and develop a temperature measurement technique applicable to typical regions of the IGM which should be able to distinguish between scenarios where reionization ends at z 6 and at z 10. ∼ ∼ Lastly, we turn our attention to 21-cm observations during reionization in 5.2 We § demonstrate that, while precise mapping of 21-cm emission from neutral hydrogen should be infeasible by first and second generation interferometers, it may be possible to make crude maps of the reionization process and identify individual ionized regions along with their approximate sizes. This measurement would be complimentary to those of the 21- cm power spectrum and would provide us with direct confirmation that we are observing reionization.

1Based on Lidz and Malloy (87). 2Based on Malloy and Lidz (97).

81 Chapter 2

How to Search for Islands of Neutral Hydrogen in the z 5.5 ∼ IGM

2.1 Introduction

It has been nearly half a century since Gunn and Peterson (65) pointed out that the lack of prominent absorption troughs, blueward of the Ly-α line in quasar spectra, implies that intergalactic hydrogen is highly ionized. Only in the year 2001 were complete “Gunn- Peterson” absorption troughs finally revealed in the Ly-α forest of high redshift (z & 6) quasars discovered using the (SDSS) (13, 46, 51). Although these prominent absorption troughs were discovered more than a decade ago, the precise interpretation of the observations, and their implications for the reionization history of the universe, remain unclear. One difficulty here relates to the large optical depth to Ly-α absorption: near z 6, the optical depth is τ 4 105 in a fully neutral IGM at the ∼ α ∼ × cosmic mean density (65). Based on this, it is common to infer that the IGM must be highly ionized below z . 6, at which point quasar spectra do show some transmission through the

82 2.1 Introduction

Ly-α line. In addition, it is clearly hard to discern whether the gas above z & 6 – that does show complete absorption in the Ly-α line – is mostly neutral or is only neutral at the level of about one part in ten-thousand or so (e.g. Fan et al. 52); in either case, the Ly-α line will be completely absorbed. However, if reionization is sufficiently inhomogeneous and ends late, there may be some transmission through the Ly-α forest before reionization completes (88, 116). Theoreti- cal models of reionization show that the IGM during reionization resembles a two-phase medium, containing a mixture of highly ionized bubbles along with mostly neutral regions. The ionized bubbles grow and merge, eventually filling essentially the entire volume of the IGM with ionized gas; the redshift at which this process completes is highly uncertain and still awaits definitive empirical constraint. In principle, the ionized bubbles may allow transmission through the Ly-α forest even when some of the IGM volume is still in fact filled by neutral regions, i.e., before reionization completes. This calls into question the conventional wisdom described above – that the presence of transmission through the z . 6 forest necessarily implies reionization completed by z = 6 (88, 116) – strictly speaking, this conclusion follows only in the unrealistic case of a homogeneously-ionized IGM. Indeed, some portions of the z 5 6 Ly-α forest are completely absorbed, while ∼ − other portions of the forest at these redshifts show transmission through the Ly-α line. Quantitatively, if one counts only the fraction of pixels with some transmission through the forest as “certain to be ionized”, the volume-averaged neutral hydrogen fraction need only be smaller than x < 0.2 at 5 z 5.5, and smaller than x < 0.5 at z = 6 h HIi ≤ ≤ h HIi (105). These constraints are conservative since even mostly-ionized gas will give rise to some completely absorbed regions at these redshifts, but it is nevertheless interesting to ask whether some of the absorbed regions could in fact come from remaining “islands” of mostly neutral hydrogen gas in the IGM. The dark pixel fraction constraints of McGreer et al. (105) certainly leave plenty of parameter space open for reionization completing at z 6. ≤

83 2.1 Introduction

In fact, there are hints – albeit indirect ones – that significant amounts of neutral gas may remain in the IGM at these late times and so we believe that investigating this possibility amounts to more than closing a remaining “loophole” in the analysis of the z . 6 Ly-α forest. For example, recent measurements of the rest-frame ultraviolet galaxy luminosity function suggest a relatively low ionizing emissivity at z & 5 6, even for − seemingly generous assumptions about the escape fraction of ionizing photons (f 0.2) esc ∼ and allowing significant extrapolations down the faint end of the luminosity function; e.g. the preferred model of Robertson et al. (156) (that matches these observations) has x = h HIi 0.1 at z = 6. In addition, the fraction of Lyman-break galaxies with detectable Ly-α emission lines shows evidence for a rapid drop between z 6 7 which may require a ∼ − significant neutral fraction at z 7 (e.g., Pentericci et al. 142, Schenker et al. 163, although ∼ see Bolton and Haehnelt 20, Taylor and Lidz 170). The inferred z 7 neutral fraction here ∼ would be easier to accommodate if there is still some neutral gas at z 6. Furthermore, ≤ Becker et al. (9) recently discovered an impressive 110 Mpc/h dark region in the z 5.7 ∼ ∼ Ly-α forest. This may result from an upward opacity fluctuation – driven by a fluctuating ultraviolet radiation field in a mostly ionized IGM – but this striking observation invites contemplating the more radical possibility that diffuse neutral regions remain in the IGM at this late time. Finally, Mesinger and Haiman (120) and Schroeder et al. (164) argue that the proximity zones of quasars at z 6 show evidence for damping wing absorption and ≥ a significant neutral fraction, further motivating the search for neutral gas at slightly later times. Perhaps more importantly, we can design robust observational tests for the presence of neutral islands in the z 5.5 IGM, and either definitively detect neutral hydrogen at ∼ these redshifts, or significantly improve on the existing upper limits from McGreer et al. (105). Towards this end, we study three possible tests for identifying neutral islands in the z 5 6 IGM, each of which can be applied using existing Ly-α forest spectra. The presence ∼ − of some transmission through the Ly-α forest at z 6 allows us to consider tests that can ≤

84 2.1 Introduction

not be applied at still higher redshift where the forest is completely absorbed (asides for in the “proximity zones” close to the quasar itself). We develop these tests using mock quasar spectra extracted from the numerical reionization simulations of McQuinn et al. (108). The first test we consider has been studied before (e.g. Fan et al. 52, Mesinger 117, McGreer et al. 105), but is the most model dependent: the abundance and size distribution of “dark gaps”, i.e., regions of saturated absorption in the Ly-α forest. Here we focus on the plausible impact of inhomogeneous reionization on the dark gap statistics. The second test utilizes the fact that the natural line width of the Ly-α line gives rise to extended damping wing absorption, in the case that highly neutral gas is present in the IGM (122). As a result, the transmission recovers more slowly around significantly neutral absorbed regions than around absorbed yet ionized regions. We find that this signature can be detected in partly neutral models by examining the stacked profile around extended absorbed regions. Note that, in contrast to previous work, here we propose to search for the damping wing signature in typical regions of the IGM, as opposed to in the proximity zones of quasars (Mesinger and Haiman 120, Schroeder et al. 164), or redward of Ly-α at the source redshift. Our third test involves the stacked profile of extended absorbed regions in the Ly-β forest. If these regions are significantly neutral, there should be a feature from absorption in the deuterium Ly-β line just blueward (but not redward) of absorbed regions. The outline of this chapter is as follows. In 2.2, we briefly discuss which range of § (volume-averaged) neutral fractions are physically plausible at z = 5.5. In 2.3 we describe § the simulations used and the process for generating mock spectra. We discuss how the dark gap size distribution may be used to constrain the neutral fraction in 2.4. In 2.5, we § § describe how quasar spectra may be stacked in order to reveal the presence of deuterium and HI damping wing absorption in an idealized scenario and discuss adapting this approach for more realistic spectra in 2.6. We then apply this approach to mock quasar spectra § in 2.7, discuss the constraining power of the stacking approaches in 2.8, and conclude § § in 2.9. Throughout, we consider a ΛCDM cosmology parametrized by n = 1, σ = 0.8, § s 8

85 2.2 Viability of Transmission Through a Partially Neutral IGM

Ωm = 0.27, ΩΛ = 0.73, Ωb = 0.046, and h = 0.7, (all symbols have their usual meanings), broadly consistent with recent Planck constraints from Ade et al. (2).

2.2 Viability of Transmission Through a Partially Neutral IGM

Ideally, this study would make use of mock Ly-α forest spectra extracted from fully self- consistent simulations of reionization, in which the efficiency of the ionizing sources and other relevant parameters are tuned so that reionization completes at z 6. Unfortunately, ≤ large-scale reionization simulations that simultaneously resolve the properties of the gas distribution, as well as the sources and sinks of ionizing photons, while capturing large enough volumes to include a representative sample of the ionized regions, are still quite challenging. Here, we instead explore more approximate, yet more flexible, models. As we describe in more detail in the next section, we make use of the reionization simulations of McQuinn et al. (108) to describe the size and spatial distribution of the ionized and neutral regions during reionization. Inside of the ionized regions, we rescale the simulated photoionization rates, adjusting the intensity of the UV radiation field to match the observed mean transmitted flux through the Ly-α forest. For simplicity, we assume that the intensity of the UV radiation field in the ionized regions is uniform and comment on the possible impact of this approximation where relevant. Before proceeding further, however, it is worth considering which (volume-averaged) neutral fractions are physically plausible near z 5.5. In order to get transmission through ∼ the z 5.5 Ly-α forest, at least some of the hydrogen needs to be highly ionized. This ∼ requires the mean free path of ionizing photons to be relatively large, although we should keep in mind that the attenuation length will vary spatially during and after reionization, and so this quantity needs to be large only across some stretches of the IGM. This in turn demands some minimum separation between the neutral islands, because otherwise the

86 2.2 Viability of Transmission Through a Partially Neutral IGM

neutral islands themselves will limit the mean free path and prevent a sufficiently intense UV radiation field from building up between the islands. Hence, it may be inconsistent to have remaining neutral islands in the IGM, yet still have some transmission through the Ly-α forest. Here we briefly quantify this reasoning; we will be content with only a rough estimate, as our focus here is more on designing empirical tests. Further theoretical exploration here might be valuable, however, perhaps along the lines of Xu et al. (187). Quantitatively, previous studies infer that a photoionization rate on the order of Γ HI ∼ 5 10−13s−1 is required to match the mean transmitted flux in the z 5.5 Ly-α for- × ∼ est (e.g., Kuhlen and Faucher-Gigu`ere 81, Bolton and Haehnelt 19).1 If we demand that the photoionization rate between the neutral islands needs to be in this ballpark to allow transmission through the forest, we can translate this into a required minimum average separation between the neutral islands, given an assumed ionizing emissivity. The average ionizing emissivity is likely on the order of ǫ 3 photons per atom per Gyr (e.g. Bolton HI ∼ and Haehnelt 19). This is close to the value required simply to balance recombinations and maintain the ionization of the IGM at the redshifts of interest. This emissivity is also comfortable with that inferred from the above photoionization rate and measurements of the mean free path to ionizing photons (Bolton and Haehnelt 19, although Becker and Bolton 7 recently argued for a slightly larger value), as well as the UV emissivity implied by measurements of the galaxy luminosity function (e.g. Robertson et al. 156). In this context, it is useful to note that: β Γ = ε σ λ , (2.1) HI HI HI,lim mfp β + 1.5 where ε is the average proper ionizing emissivity, σ = 6.3 10−18cm2 is the photoion- HI HI,lim × ization cross section at the Lyman limit, λmfp is the mean free path of ionizing photons at the Lyman limit, and β is the intrinsic, unhardened spectral index of the ionizing radiation.

1These studies assume that reionization is complete at these redshifts. If the universe is in fact partly neutral, then a higher photoionization rate should be required in the ionized regions. In our rough estimate here, we neglect this given the other significant uncertainties involved.

87 2.2 Viability of Transmission Through a Partially Neutral IGM

This expression assumes that the mean free path to ionizing photons propagating through a clumpy IGM scales as ν3/2 (198). Inserting typical numbers we find:

ε Γ = 5.0 10−13sec−1 HI (2.2) HI × 3 photons/atom/Gyr   β 3.5 1+ z 3 λ mfp . (2.3) × 2 1.5+ β 6.5 9.1 pMpc       In other words, to get transmission through the forest for plausible values of the ionizing

emissivity, we require the mean separation between neutral islands to be λmin & λmfp & 9.1pMpc. This is a minimal requirement in that it assumes the neutral islands set the mean free path, when in fact Lyman limit systems and cumulative absorption in the mostly ionized gas may also play a role. On the other hand, the required minimum separation between the

neutral islands would go down if a smaller ΓHI suffices to allow transmission through the forest, or if the ionizing emissivity is in fact higher. However, the mean free path to ionizing photons has recently been measured at z = 5.16 to be λ = 10.3 1.6pMpc (183), only mfp ± somewhat larger than our assumed λmfp here. While there are still uncertainties, and while the measured mean free path scales steeply with redshift (λ (1 + z)5.5), viable models mfp ∝ are unlikely to have neutral islands spaced much more closely than this.

We can then use this requirement on λmfp to get some sense of which volume-averaged neutral fractions are plausible at z 5.5. In the simulation outputs considered here (see ∼ 2.3), the mean separation between neutral islands is λ = 17.0 pMpc, 5.3 pMpc, and § mfp 2.7 pMpc for x = 0.05, 0.22, and 0.35, respectively. The first case certainly satisfies h HIi the requirement described above, the second case is just a bit on the small side, while the third case is uncomfortably small. Given the uncertainties in this argument, and the possibility that the neutral islands are a bit larger than in our simulation (which would increase their mean separation at fixed filling factor), we consider all three cases, but refrain from considering still more neutral models. We regard the latter case ( x = 0.35) as an h HIi extreme scenario intended mostly for illustration.

88 2.3 Simulations and Mock Spectra

Finally, it is worth keeping in mind that any remaining neutral islands will likely be photoionized on a short timescale. For example, using Eq. 1 in Lidz and Malloy (87) with 9 C = 3, Mmin = 10 M⊙, and ζ = 20, the redshift interval over which the volume average ionized fraction transitions from x = 0.8 to x = 1 is only ∆z 0.5. However, it is h ii h ii ∼ possible that we are catching this – likely brief – phase in z 5.5 Ly-α forest spectra and ∼ the possibility of testing this remains tantalizing.

2.3 Simulations and Mock Spectra

With the above discussion to frame the range of possibilities, we move to describe the numerical simulations used in this analysis and our approach to constructing mock Ly- α forest absorption spectra before reionization completes. We use simulated density and ionization fields generated from a dark matter simulation of McQuinn et al. (108) which tracks 10243 dark matter particles in a simulation volume with a co-moving sidelength of L = 130 Mpc/h. We assume that the gas closely follows the dark matter. In this work, we focus on redshift z = 5.5, but consider several possible neutral fractions. In practice, we obtain ionization fields with higher (lower) neutral fractions by using simulation outputs at higher (lower) redshifts. This should be an appropriate approximation since the statistical properties of ionized regions at a given neutral fraction are most sensitive to the neutral fraction and are relatively insensitive to the redshift at which the neutral fraction was attained (see McQuinn et al. 110 and Furlanetto et al. 62). We generate mock quasar spectra according to the usual “fluctuating Gunn-Peterson” approach (e.g., Croft et al. 39), with a few refinements to capture the main effects of incomplete reionization. First and foremost, we do not assume a fully ionized IGM. The transmission in the Ly-α forest is sensitive to the precise ionized fractions in the ionized phase of the IGM. In order to simplify our study, as mentioned in the previous section, we rescale the simulated photoionization rates in the ionized regions to match the observed mean transmitted flux through the Ly-α forest. We do this assuming ionization equilibrium,

89 2.3 Simulations and Mock Spectra

and a constant value of the UV background (with a photoionization rate per atom of ΓHI)

within ionized regions. Specifically, simulated pixels with xi > 0.9 are considered highly ionized while less ionized pixels are considered fully neutral.1 This simplified approach allows us to consider a range of different possibilities for the ionization state of the IGM quickly. We comment on the shortcomings of this approach when appropriate. The optical depth of a given pixel, i, in the simulation can then be found by summing over contributions from neighboring pixels (Bolton & Haehnelt 2008):

cσαδR nHI(j) τα(i)= H(a, x), (2.4) π1/2 b(j) Xj 1/2 where b(j) = (2kBT (j)/mp) is the Doppler parameter, T (j) is the temperature of pixel j, δR is the pixel proper width, σ = 4.48 10−18cm2 is the Ly α scattering cross section, α × mp is the proton mass, H(a, x) is the Hjerting function, and nHI(i) is the number density of hydrogen atoms at pixel i, found using the simulated density field. To calculate the Doppler parameter, we assume that the gas obeys a modified temperature-density relationship

T (1 + δ)γ−1 if ionized T (δ)= 0 (2.5) (1, 000K if neutral, where δ is the matter overdensity in units of the cosmic mean and we choose T = 2 104K 0 × and γ = 1.3 as the temperature at mean density and slope of the temperature-density re- lation, respectively. For simplicity, we assume the ionized gas lies on the aforementioned temperature-density relation, although there should be significant scatter around this re- lation close to reionization (e.g. Lidz and Malloy 87). We do not expect this to impact our conclusions significantly. The neutral gas should be colder than the ionized gas, of course, with a temperature set perhaps by low levels of X-ray pre-heating. Here we adopt

1After effectively thresholding the ionization field in this way, we end up with neutral fractions which are ≈ 20% higher than in the original simulation. Throughout the chapter, we refer to increased, thresholded neutral fractions.

90 2.3 Simulations and Mock Spectra

T = 1, 000 K for the neutral gas; this choice is likely a bit large (it was chosen partly for ease in computing the Hjerting function below), but we have checked that we get nearly identical results for colder temperature choices. The Hjerting function is a convolution of a Lorentzian profile, which incorporates the natural line profile of the Lyman-series lines, with a Maxwell-Boltzmann distribution, which accounts for the effects of thermal broadening on the line profile. The Hjerting function is defined by:

2 a ∞ e−y dy H(a, x)= , π a2 + (x y)2 Z−∞ − where a = Λ λ /4πb(j), Λ = 6.265 108 sec−1 is the damping constant, λ = 1215.67A˚ is α α α × α the Ly α wavelength, x is the relative velocity of pixel i and pixel j in units of the Doppler parameter, defined as x = [v (i) u(j)] /b(j), where u(j) = v (j)+ v (j). The peculiar H − H pec velocity field is generated by applying linear perturbation theory to the underlying density field.1 In detail, the natural line profile is only approximately described by a Lorentzian (83), with asymmetric corrections becoming important far from line center. In this study, the precise shape of the damping wing far from line center is unimportant: we make use only of the gradual recovery in transmission around saturated neutral regions, rather than the detailed shape of this recovery, which is also strongly influenced by neighboring neutral regions. We hence expect the Lorentzian form to be a good approximation for our present purposes. In addition to including absorption from the hydrogen damping wing, we also include absorption from primordial deuterium. As a result of big bang nucleosynthesis, primordial hydrogen should be accompanied by traces of deuterium, with a relative abundance by number of D/H = 2.5 10−5 (Cooke et al. 37). Due to its slightly increased reduced × mass, transitions in deuterium will be shifted blueward by 82km/s compared

1This was done because the full peculiar velocity field was not readily available, but this approximation should not impact our results.

91 2.3 Simulations and Mock Spectra

to the same transitions in hydrogen. We account for deuterium by scaling the number density of hydrogen in a given pixel by the relative abundance and shifting the resulting optical depths blueward by 82km/s. Additionally, the Doppler parameter is adjusted to 1/2 bD(j) = (2kBT (j)/(2mp)) to account for the increase in mass. In this work, we focus mostly on z = 5.5 and adopt a mean transmitted flux at this redshift of F = 0.1, consistent with determinations from e.g., Becker et al. (13). In some h i cases, we test the sensitivity of our results to the mean transmitted flux by considering F = 0.05 as well. In general, the lower the mean transmitted flux, the more challenging it h i is for us to identify any remaining neutral islands. On the other hand, the likelihood that neutral islands remain increases towards high redshift and decreasing mean transmitted flux. As mentioned previously, we rescale the simulated photoionization rates in the ionized regions to a uniform value, normalized so that an ensemble of mock spectra matches the observed mean transmitted flux. It is important to note that the mean transmitted flux is a very steep function of redshift near z 5.5, and that the sightline-to-sightline scatter in ∼ this quantity is substantial (52), and so one may want to carefully test for sensitivity to the precise redshift binning used. We use the same approach as described above to generate Ly β mock spectra, with λ = 1025.72A,˚ Λ = 1.897 108 sec−1, and σ = 7.18 10−19cm2. However, in generating β β × β × Ly β mock spectra, we must also account for foreground Ly α absorption due to gas at lower redshifts, λα(1 + zLyα) = λβ(1 + zLyβ), where zLyα is the redshift of the foreground Ly α

absorber and zLyβ is the redshift of the Ly β absorber. We will assume we are investigating

quasar spectra at zLyβ = 5.5 for this work, such that the corresponding foreground Ly α ab-

sorption in the Ly β spectra occurs at redshift zLyα = 4.5. We adjust ΓHI for the foreground Ly α absorption to match measurements of the mean transmission from Becker et al. (11) at these redshifts ( F 0.31 at z = 4.5).1 The optical depth of a pixel in a Ly β spectrum h i≈ 1We generate foreground Ly α absorption by considering the absorption from regions in the same simula- tion box, but demand that they are widely-separated from the high redshift regions of interest (> 10 Mpc/h). This enforces that the underlying density fields sourcing the Ly β absorption and the foreground Ly α ab-

92 2.3 Simulations and Mock Spectra

is then the sum of the contribution from the foreground Ly α absorption and the intrinsic tot Ly β absorption τβ (zLyβ)= τβ(zLyβ)+ τα(zLyα). In Fig. 2.1, we show an example mock Ly α spectrum for a particular line of sight through the simulation. We show only a portion of the line of sight in order to exhibit smaller-scale features. The top figure shows the Ly α transmission when the hydrogen damping wing is neglected (black) and when it is included (dashed red), while the bottom panel shows the underlying thresholded ionization field. We have neglected peculiar velocities in creating this figure in order to facilitate a comparison between the spectrum and the underlying ionization field. From this figure, we see that the damping wing indeed has a significant effect on the transmission, but that its effect is hard to discern without knowing the damping-wing-less transmission. This is the case for two reasons. First, the forest here is very absorbed and the damping wing absorption becomes mixed with resonant absorption from neighboring ionized regions. Second, the damping wing from a particular neutral region may overlap with the damping wing from another neutral region, altering the shape of the resulting absorption. Specifically, we see that, in the example spectra, the region at v 4500 km/s ≈ is sandwiched between HI regions to the left and right, both within 1000km/s. Therefore, the optical depths for the corresponding pixels likely have significant contributions from resonant absorption, damping wing absorption from the HI region to the left, and damping wing absorption from the HI region to the right. While detecting individual instances of damping wing absorption in this case seems impossible, we will show that detecting the presence of damping wing absorption on average should be feasible through the stacking of high-redshift quasar spectra. sorption are uncorrelated, as should be the case for actual spectra.

93 2.4 Dark Gap Statistics

1 No HI Damping Wing w/ HI Damping Wing 0.8

0.6 F 0.4

0.2

0 0 1000 2000 3000 4000 5000 6000 v (km/s)

1

0.8

0.6 HI x 0.4

0.2

0 0 1000 2000 3000 4000 5000 6000 v (km/s)

Figure 2.1: Example mock Ly α forest spectrum and corresponding neutral fraction. The top panel shows the Ly α transmission while the bottom panel is the neutral fraction along the line

of sight, with ionized regions set to xHI 0 for illustration. The black curve in the top panel ≈ shows the transmission through the forest when absorption due to the hydrogen damping wing is neglected, while the red curve includes damping wing absorption. The comparison illustrates that damping wing absorption has a prominent impact, but it is also clear that the presence of the damping wing will be hard to discern by eye. The line of sight is extracted from a model

with xHI = 0.22, but note that we have deliberately chosen a sightline with more neutral h i regions than typical.

2.4 Dark Gap Statistics

With the mock spectra of the previous section in hand, we now consider the size distribution of regions of saturated absorption – dark gaps – and its dependence on the underlying neutral fraction. Using such dark gap statistics has been widely discussed as a potential

94 2.4 Dark Gap Statistics

probe of the IGM neutral fraction (see, e.g., Fan et al. 52, Gallerani et al. 63, Mesinger 117, McGreer et al. 105). In a fully ionized IGM, the size of dark gaps in quasar spectra should grow with increasing redshift, simply owing to the increasing mean density of the universe and as a result of any decline in the intensity of the UV radiation background. However, once quasar spectra start to probe the tail end of reionization, the increase in dark gap size should accelerate due to the presence of islands of neutral hydrogen. In Fig. 2.2, we have plotted the size distribution of dark gaps, dP (L)/d ln L, in blue for the x = 0.22 mock spectra, assuming a mean transmission of F = 0.1. Additionally, h HIi h i the dashed curves display the two underlying populations of dark gaps: those sourced by ionized gas (magenta) and those sourced by neutral gas (cyan). For clarity, we only show dark gaps larger than L = 0.7 Mpc/h ( 90km/s), since smaller saturated regions sat,min ∼ will be predominantly ionized. Additionally, we have neglected peculiar velocities when generating spectra here. Two important points become apparent from this figure. First, at L 8.5 Mpc/h ( 1100km/s), dark gaps transition from being primarily sourced by ∼ ∼ ionized gas to being primarily sourced by neutral gas. This reinforces our intuition that, in a partially neutral IGM, the largest dark gaps should correspond to the remaining neutral islands. Second, the dark gaps being composed of two different populations gives rise to a bimodality in the size distribution. This suggests that the behavior of the large-L tail of the size distribution may offer additional information about the neutral fraction, with a steep decline suggesting a highly ionized IGM and a more gradual decline, or the emergence of a second peak, suggesting a significantly neutral IGM. Such a “knee” in the dark gap size distribution is also mentioned in Mesinger (117). Additionally, we can consider the large-L tail of the size distribution and its dependence on neutral fraction at a fixed mean transmission. In Fig. 2.3, we plot an expected histogram of dark gap sizes for 10 spectra for x = 0 (magenta), 0.05 (cyan), 0.22 (blue), and 0.35 h HIi (black), again assuming that F = 0.1. Three trends become obvious from this plot. First, h i as the neutral fraction is increased (at fixed F ), the number of large saturated regions h i

95 2.4 Dark Gap Statistics

increases and, second, as the neutral fraction is increased, the size of the largest dark gaps also increases. For example, in the x = 0.22 model, the largest dark gaps are roughly h HIi five times bigger than in the fully ionized model. Additionally, we again see hints of the underlying dark gap size distribution being bimodal as the neutral fraction is increased, supporting the idea that the shape of the dark gap size distribution may be a diagnostic for the underlying neutral fraction. Given these trends, it should be possible to compare dark gap distributions from ob- served spectra against models at various neutral fractions and use this to constrain the mean neutral fraction of the IGM. This approach is appealing in that it does not require especially high-resolution or high signal-to-noise spectra. However, it does require comparison with simulated models of the dark gap size distribution and so the conclusions reached will be somewhat model dependent. Additionally, the distributions are dependent on the assumed mean transmission, which is itself uncertain. In particular, estimates of the mean trans- mission at high redshift may be impacted by continuum fitting errors, given the inherent difficulty in estimating the unabsorbed continuum level in highly-absorbed spectra. In order to investigate the impact of possible continuum fitting errors, we generate mock spectra in the fully ionized model with F = 0.03 but then rescale the flux in each simulated h i pixel by a multiplicative factor – to mimic the effect of continuum misplacement – such that the measured mean transmitted flux appears to be F = 0.1. This case is shown as the h measi magenta dashed line in Fig. 2.3. Here the dark gap size distribution is shifted towards sizes than one would expect in an ionized model at F = F = 0.1. However, the shape of h i h measi the size distribution is still quite different than in the partly neutral models. Importantly, the dark gap distribution in the ionized model still lacks the distinctive bump at large sizes that is the hallmark of a partly neutral IGM in our models.

96 2.5 Stacking Toy Spectra

0 10 HI+HII Sat. Regions HII Sat. Regions HI Sat. Regions −1 10

−2

dP(L)/dlnL 10

−3 10

0 1 10 10 L (Mpc/h)

Figure 2.2: Dark gap size distribution for the xHI =0.22, F =0.1 model. The solid blue h i h i curve shows the total distribution of dark gaps from an ensemble of mock spectra, where the magenta (cyan) curve shows the same thing but for the dark gaps sourced by ionized (neutral) gas. Here, we have focused on dark gaps with L> 0.75 Mpc/h. This clearly demonstrates that neutral hydrogen is the dominant source of large dark gaps in our mock spectra, provided there is an appreciable neutral fraction.

2.5 Stacking Toy Spectra

In this section we describe our basic approach of stacking Ly α and Ly β spectra in order to detect the presence of HI damping wings and absorption due to deuterium, respectively. While the forest is too absorbed at these redshifts to easily detect damping wings or deu- terium absorption due to individual neutral regions, here we demonstrate that the presence of such features can be revealed on average by stacking regions of transmission over many spectra. This section serves as a proof of principle by applying a simplified stacking approach to mock spectra generated using an idealized IGM model. Specifically, we consider an ensemble of sightlines through our simulation box and assume that the IGM is entirely ionized with the exception of a single HI island with mean density and varying length, L,

97 2.5 Stacking Toy Spectra

2 10

1 10

= 0.35 HI = 0.22

Scaled Frequency HI

0 = 0.05 10 HI = 0 HI

0 1 2 10 10 10 L (Mpc/h)

Figure 2.3: Large-length tail of the dark gap size histogram for xHI = 0 (magenta), 0.05 h i (cyan), 0.22 (blue), and 0.35 (black) for the case when F = 0.1. The y-axis is scaled to h i indicate the expected number of dark gaps obtainable from 20 spectra. Bins in this figure are spaced logarithmically. The dashed magenta line indicates the dark-gap size distribution in the fully ionized case when the true transmission is F =0.03, but continuum fitting errors result h i in a measured mean transmission of Fmeas =0.1. h i inserted randomly along each line of sight. We then generate mock spectra assuming these density and ionization fields. The stacking in this section is always done starting at the HI/HII boundaries of a given HI region moving outward.

2.5.1 HI Damping Wing

Our stacking approach can be clearly demonstrated by considering the damping wing from neutral hydrogen. Due to the natural width of the Ly α line, a neutral hydrogen gas parcel should cause Ly α absorption over a range of frequencies. Far from line center, this

98 2.5 Stacking Toy Spectra

absorption will have an optical depth roughly following (Miralda-Escude and Rees 125):

τ R c 1 1 τ DW(∆v) GP α (2.6) Lyα ≈ π ∆v − ∆v + v  ext  where τ is the Gunn-Paterson optical depth, R Γ λ /4πc, Γ = 6.265 108 sec−1 is GP α ≡ α α α × the Ly α decay constant, ∆v is the separation from the HI/HII boundary in velocity space,

vext is the extent of the hydrogen region in velocity space, and c is the speed of light. For a large neutral region, this equation implies that τ DW( v < 600km/s) 1 at z 5.5. This Lyα | | ≥ ∼ excess absorption is referred to as the hydrogen “damping wing”. While both neutral gas and highly ionized gas can cause absorption in quasar spectra, only a significantly neutral hydrogen patch will result in damping wing absorption, owing to the greatly reduced optical depth in the wing compared to line center. As such, detecting damping wing absorption would be a smoking gun for the presence of significantly neutral hydrogen islands. Note that the transmission profile will differ from the simple form of Eq. 2.6, owing mostly to neighboring neutral regions, however the gradual recovery to transmission around saturated neutral regions should be a distinctive indicator that highly neutral regions remain in the IGM. In Fig. 2.4 we show the results of stacking transmission outside of neutral regions in the toy mock spectra described earlier in this section, neglecting deuterium for the time being. Namely, we show the stacked transmission outside neutral islands of length L = 0.76 Mpc/h ( 100km/s) in black, L = 1.27 Mpc/h ( 170km/s) in blue, L = 5.34 Mpc/h ( 700km/s) ∼ ∼ ∼ in cyan, and stacked transmission neglecting the damping wing in red. Additionally, we have plotted the analytic curves corresponding to Eq. 2.6 for the various L values, shown with dashed curves. We have applied a single multiplicative factor to these curves to account for average resonant absorption from ionized gas. Together, this figure implies that damping wing absorption from isolated neutral regions has a significant impact on quasar spectra, extending 1000km/s past the HI/HII boundaries, which may be observable through ∼ stacking as expected from Eq. 2.6.

99 2.5 Stacking Toy Spectra

In providing a toy example of how the hydrogen damping wing can affect spectra, we have neglected many important challenges that such a measurement would face. For example, we assumed perfect knowledge of the underlying ionization state of the IGM in order to determine where to stack and we assumed that we could discriminate between neutral and highly ionized absorption systems. However, the presence of such a large and potentially-observable feature provides motivation for us to apply the stacking approach in a more realistic manner. In 2.6 and in Appendix B, we describe several such challenges § and subtleties along with potential resolutions.

0.1

0.08

0.06 Transmission L = 0.76 Mpc/h α 0.04 L = 1.27 Mpc/h L = 5.34 Mpc/h 0.02 No Damping Wing Stacked Ly

0 0 200 400 600 800 1000 1200 ∆ v (km/s)

Figure 2.4: Stacking idealized Ly α spectra containing toy HI regions. The above figure shows the stacked transmission outside isolated HI regions with mean density and size L =

0.76 Mpc/h (vext 100km/s), L = 1.27 Mpc/h (vext 170km/s), and L = 5.34 Mpc/h ≈ ≈ (vext 700km/s) shown in black, blue, and cyan, respectively. The solid red curve shows the ≈ stacked transmission outside of the same HI regions neglecting the damping wing, which will be the same on average in all cases. In generating these spectra, we assume F = 0.1. In h i this greatly-idealized case, the presence of the hydrogen damping wing is seen clearly through extended excess absorption compared to the red curve. Furthermore, we can see that the excess absorption closely follows what we would expect analytically based on multiplying Eq. 2.6 by the overall mean transmission. In this figure, all stacking starts at HI/HII boundaries.

100 2.5 Stacking Toy Spectra

2.5.2 Deuterium

With the stacking approach of the previous section in mind, we now consider absorption due to deuterium. As noted in 2.3, primordial hydrogen should be accompanied by traces § of deuterium, with a relative abundance of 2.5 10−5 (Cooke et al. 37). Due to its slightly ∼ × increased reduced mass, atomic transitions in deuterium are shifted blueward by 82km/s compared to the same transitions in hydrogen. This implies that absorption due to neutral hydrogen in the IGM should be accompanied by additional absorption from deuterium, shifted blueward by 82km/s. We can estimate the optical depth for Ly α absorption in deuterium at cosmic mean density by simply scaling the hydrogen Ly α optical depth by the deuterium abundance:

D 1+ z 3/2 τ = τ 8.25x (1 + δ) . (2.7) D,α H × GP ≈ HI 6.5     Thus, we see that while the relative abundance of deuterium is extremely small, the Gunn- Peterson optical depth is so large that the resulting deuterium optical depth is still of order 10 in Ly α. An appealing aspect of searching for damping wing absorption is that the optical depth in the wing is large enough to cause significant absorption in the presence of neutral islands, but small enough to be negligible for ionized absorption systems. We see this again in the case of deuterium absorption, suggesting that it may be useful as an additional “smoking gun” indicator for underlying neutral hydrogen. However, an obvious problem with detecting deuterium in Ly α spectra is that the feature should be narrow and well within the broad range of velocities where the hydrogen damping wing is significant. Specifically, according to Eq. 2.6, at ∆v = 82km/s, the damping wing optical depth for an extended neutral region should be τ DW(∆v 82km/s) 8. Therefore, the feature should be completely wiped out Lyα ≈ ≈ in Ly α spectra by the hydrogen damping wing.

101 2.5 Stacking Toy Spectra

However, the damping wing optical depth in the Ly β line is much smaller. Specifically, according to Eq. 2.6, the damping wing optical depth scales as

DW τLyβ τGP,β Rβ fβλβ (Γβ +ΓHα)λβ DW = = τLyα τGP,α × Rα fαλα × Γαλα 2 2 fβ fHα λβ = 2 1+ 2 = .0410, (2.8) fα fβ λHα !

and should therefore be significantly narrower in Ly β than in Ly α. In the above expression, fα, fβ, and fHa are the oscillator strengths of the Ly α, Ly β, and Balmer-α transitions, respectively, with λ denoting the corresponding wavelengths and Γ denoting the correspond- ing decay constants. By modifying Eq. 2.6 for Ly β, we see that τ DW( ∆v & 25km/s) 1. Lyβ | | ≤ Therefore we find that the hydrogen damping wing should not wipe out deuterium absorption features in Ly β. Furthermore, while the hydrogen damping wing optical depth is reduced 2 2 by a factor of roughly fβ/fα when considering Ly β, the total optical depth in the deuterium line is only reduced relative to deuterium Ly α by f λ /f λ 1/6, such that the optical β β α α ≈ depth should still be of order 1 for deuterium Ly β. Therefore, not only should a deuterium absorption feature survive the hydrogen damping wing, but it should still have a strong enough optical depth to cause significant absorption if neutral islands in fact remain. Naturally, it should be very difficult to detect individual deuterium absorption features from the diffuse IGM, as the Ly-β spectra will be very absorbed when the universe is neutral enough to produce the features in the first place. However, the feature may nonetheless be observable on average through the stacking of high-resolution quasar spectra. In order to demonstrate the strength of the deuterium absorption feature in stacked spectra, we incorporate deuterium into the same toy sightlines from 2.5.1 to produce mock Ly β spec- § tra, neglecting foreground Ly α absorption for the time being. We are then able to stack transmission outside of neutral regions in the spectra, starting at the HI/HII boundaries and moving outward. However, since deuterium absorption will only occur on the blue side of neutral regions, we need only stack those regions of transmission. In fact, this offers a

102 2.5 Stacking Toy Spectra

clean test for detecting deuterium. Namely, we can separately stack transmission redward and blueward of neutral regions and compare. Excess absorption on the blue side of neutral regions, on average, could signal the presence of deuterium absorption. This is especially appealing since there should be no sources of contamination that would cause a similar, and significant, red/blue asymmetry.1 In Fig. 2.5, we show the results of stacking transmission in Ly β redward (red) and blue- ward (black) of the toy neutral regions across the full ensemble of mock quasar spectra. As in Fig. 2.4, all stacking begins at HI/HII boundaries. We can see the blueward transmission clearly exhibits excess absorption due to deuterium extending roughly 80km/s from the ∼ HI/HII boundary. Thus, in this idealized scenario, the presence of deuterium in islands of neutral hydrogen leaves a very clear signature in the stacked Ly β transmission. Before proceeding further, we should point out one important caveat here. In our simulated models, the transition between fully neutral and highly ionized regions is, by construction, perfectly sharp. If this transition is more gradual in reality, then the narrow deuterium feature could be overwhelmed by absorption from mostly ionized hydrogen in this transition region. A minimal scale for this transition region is set roughly by the mean free path to ionizing photons through the neutral IGM, which is only λ 1/(n σ ) HI ∼ HI HI ≈ 6proper kpc/h 0.8km/s. This minimal scale is two orders of magnitude smaller than the ≈ scale of the deuterium feature and hence does not present a worry. However, if the edges of the ionized regions tend to experience a reduced ionizing background, this might obscure the deuterium feature, even in the case of a partly neutral IGM. We believe the possibility of detecting this deuterium feature is enticing enough to warrant further investigation.

1One source of asymmetry we do find, which can be seen in Fig. 2.9, results from the fact that, when dealing with realistic spectra, we force there to be transmission in Ly β at the locations where stacking begins. This results in a small selection effect, where selected neutral absorption systems have a reduced probability of having nearby neutral regions and have correspondingly-smaller nearby optical depths. For deuterium, this smaller optical depth is shifted blueward, causing less absorption on the blue side of the line for ∆v & 82km/s. However, this asymmetry is minor and opposes the asymmetry from deuterium absorption.

103 2.6 Steps of Approach

As was the case in 2.5.1, we have made several simplifying assumptions and have § additionally neglected foreground Ly α absorption from the lower-redshift IGM. However, the clear presence of deuterium absorption revealed through the simplified stacking approach provides motivation to also consider applying it to more realistic spectra, as will be discussed in 2.6. §

0.15

0.1 Blueward Redward Transmission β

0.05 Stacked Ly

0 0 50 100 150 200 ∆ v (km/s)

Figure 2.5: Presence of deuterium absorption revealed through stacking idealized Ly β spectra containing toy neutral regions. The red and black curves show the stacked Ly β transmission redward and blueward, respectively, of toy neutral regions of length L = 5 Mpc/h ( 700km/s) ≈ randomly inserted into many sightlines, with spectra generated assuming FLy =0.1. In each h αi case, stacking begins at the underlying HI/HII boundary. We have also mimicked the effect of including foreground Ly α absorption by scaling the feature by the mean transmission in the foreground Ly α forest. This demonstrates that, at least in this idealized case, the presence of deuterium absorption can be easily seen out to 80km/s past the HI/HII boundaries. ∼

2.6 Steps of Approach

In 2.5, we demonstrated the utility of stacking idealized quasar spectra in order to reveal the § presence of the HI damping wing and deuterium absorption. The success of this approach in the toy case provides motivation for us to apply it to realistic mock spectra. In doing so, we must confront the simplifying assumptions made in 2.5. §

104 2.6 Steps of Approach

The most obviously unrealistic assumption made in 2.5 is that we can precisely identify § the HI/HII boundaries underlying our spectra. In practice, we will only have access to the level of transmission at each point along the spectra. However, based on Fig. 2.5, the recovery from saturated absorption to transmission occurs within . 15km/s in Ly β from the edge of the neutral zone, and should therefore provide a relatively good indicator of the HI/HII boundary. Therefore, we choose to identify stacking locations based on where transmission recovers in Ly β. To be clear, for the case of the hydrogen damping wing, we are stacking transmission in the Ly α forest, but we are choosing where to start the stacking based on features in the Ly β forest. A drawback of this approach, when searching for the hydrogen damping wing, is that we are only able to stack regions of the Ly α forest with corresponding regions in the Ly β forest that are not contaminated by Ly γ absorption. This effectively reduces the amount of usable spectra, since, for a quasar at z = 5.5, the pure Ly α forest will extend 4.5 z 5.5, but Ly γ absorption will contaminate the Ly β forest ≤ ≤ at z . 5.16. If presented with a limited number of spectra, it may be worth searching for the damping wing by using only the Ly α regions of the spectra. By stacking at the precise locations of HI/HII boundaries in 2.5, we were also en- § suring that our sample of absorption systems was all neutral. However, when we modify our approach to begin stacking at locations where transmission recovers from saturated absorption, we may start stacking transmission outside of ionized absorption systems to- gether with transmission outside of neutral absorption systems, diluting our signal. Since the signal we are aiming to find is small to begin with, it is important that we minimize this contamination from ionized regions. To do this, we take advantage of the main argument of 2.4, namely that regions of saturated absorption sourced by neutral gas should be signifi- § cantly larger, on average, than those sourced by ionized gas. Therefore, we choose to stack only transmission outside of large regions of saturated absorption. Furthermore, since true neutral regions should cause saturated absorption in Ly β, we choose to stack only outside of large saturated regions which are fully absorbed in Ly β, where we define “large” to be

105 2.6 Steps of Approach

> 500km/s (& 4 Mpc/h) in Ly β. Note that this choice is tuned for the case of F = 0.1: h i a different choice may be better for other values of the mean transmitted flux. At any rate, in applying these tests to real data, one would likely vary this size scale across a range of possible values. Additionally, an appealing feature of the search for deuterium absorption is that it offers a very clean test for its detection, namely a red/blue asymmetry in the transmission outside of plausibly neutral regions. The disparity in the size distribution of saturated regions sourced by neutral and ionized gas suggests a similar test may be possible for the detection of the HI damping wing. Namely, while large regions of saturated absorption are likely to be sourced by neutral gas, small regions of saturated absorption are likely to be sourced by ionized gas. Therefore, to find evidence of excess absorption outside of neutral regions due to the HI damping wing, we compare the stacked transmission outside of large absorption systems, plausibly sourced by neutral gas, to that outside of small absorption systems, likely sourced by ionized gas. A significant amount of excess absorption outside of the former compared to the latter, extending further than any possible density correlations, would suggest the presence of damping wing absorption. Furthermore, in 2.5.2, we discussed how the damping wing is greatly weakened in § Ly β compared to in Ly α. Therefore, an additional test for the presence of damping wing absorption could be to take the ratio of the stacked Ly β transmission to the stacked Ly α transmission, where stacking occurs in the same physical regions in both cases. In the event that there is significant damping wing absorption, this ratio should also slowly recover to some constant value at large ∆v. We further discuss and develop this approach in Appendix B. When dealing with realistic spectra, we must adjust our approach to accommodate the presence of noise (and finite spectral resolution). While noise should average out in stacked regions, the presence of noise will also obfuscate the precise boundaries between saturated absorption and transmission. We choose to handle this by smoothing our noisy spectra over

106 2.7 Results

a scale of 100 km/s ( 0.75 Mpc/h) and defining any pixel, i, with transmission F < 3˜σ ∼ i N to be consistent with saturated absorption, whereσ ˜N denotes the standard deviation of the smoothed noise. We then define regions in the smoothed spectra where the flux goes from

F < 3˜σN to F > 3˜σN as the transitions from saturated absorption to transmission, and therefore as potential points to start stacking. When stacking transmission, however, we stack the transmission in the unsmoothed spectra. Another concern is that damping wing absorption sourced by DLAs may erroneously be attributed to a significantly neutral IGM. However, in Appendix A, we estimate the expected rate of DLAs occurring in z 5.5 quasar spectra and find it is small enough to ∼ be ignored. Additionally, DLAs may be discriminated from diffuse neutral islands based on the presence of metal lines and the relative sizes of their absorption in Ly α and Ly β. Finally, as mentioned previously, we approximate the ionizing background in the ionized regions as uniform and ignore scatter in the temperature density relation. Accounting for these fluctuations might lead to a more gradual recovery in the transmission around absorbed regions – in the case of a fully ionized universe – than in our models. Further investigation of this issue would certainly be required if a gradual recovery is indeed found in real spectra. In Appendix B, we discuss a possible empirical test that may help in this regard.

2.7 Results

Having considered the subtleties of the previous section, we are now ready to apply the three-pronged approach to more realistic mock spectra. In each section, we first consider the ideal case where no noise has been applied to give an idea of the potential constraining power of the different methods. Subsequently, we add realistic levels of noise and consider realistic spectra resolution to give an idea of the constraining power of the approaches applied to Keck HIRES spectra for the deuterium feature and damping wing, and spectra with slightly higher resolution than SDSS for the dark gap size distribution.

107 2.7 Results

2.7.1 Detecting the Damping Wing

We first consider the ability to uncover the presence of the hydrogen damping wing by strategically stacking regions of transmission in the the Ly α forest of z 5.5 noiseless ≈ mock quasar spectra. As discussed in 2.6, our aim is to compare the average transmission § outside of plausibly neutral absorption systems to the transmission outside of likely ionized systems. We identify the plausibly neutral absorption systems by requiring the regions be com- pletely absorbed in Ly β, and also that the regions of saturated absorption are at least L > 500km/s ( 4 Mpc/h) in Ly β. We begin stacking at the point in the Ly α spectrum sat ∼ which corresponds to the recovery from absorption to transmission in Ly β. We identify the likely ionized absorption systems by requiring that they are below a maximum length

Lmax = 300km/s in Ly β. In Fig. 2.6, we show the results of applying this approach to realistic mock spectra gen- erated assuming various ionization states of the IGM. In the top panel, we show the stacked transmission outside of plausibly neutral absorption systems (solid) and likely ionized ab- sorption systems (dashed), using a volume-averaged neutral fraction of x = 0.35 (black), h HIi 0.22 (blue), 0.05 (cyan), and x = 0 (magenta). The curves agree with our expectations, h HIi namely that transmission outside of neutral regions should recover more slowly and exhibit a rough damping wing shape with a large extent in velocity space. We see that the excess absorption extends farther than the . 1000km/s expected from an isolated damping wing. However, as discussed in Appendix C, we find that the spatial clustering of neutral regions is responsible for this effect. While, for several reasons discussed earlier, the shape of the absorption is distorted compared to Fig. 2.4, it can be seen for all significantly neutral ionization states. An important check is to apply the stacking procedure to a fully ionized IGM and ensure that we do not make a false detection. The results of this check are shown by the magenta curves in Fig. 2.6. As we can see, the resulting stacked transmission outside of plausibly

108 2.7 Results

neutral regions lacks an overall damping wing shape and stays roughly fixed near the mean transmission. We can also see that the transmission outside of small absorption systems is very sen- sitive to the underlying neutral fraction. We expect this, however, since this stacked trans- mission depends strongly on the average transmission in regions which are not in saturated absorption, denoted F F > 0 . Since the dark pixel covering fraction in our mock spectra h | i increases with x , mock spectra with larger neutral fractions must have larger values for h HIi F F > 0 to maintain F = 0.1. As such, Fig. 2.6 shows that the stacked transmission h | i h i outside of small absorption systems increases monotonically with x . h HIi We estimated the stacked transmission from a large ensemble of simulated spectra to produce a smooth estimate of the average transmission around saturated regions in each model. The transmission curves outside of individual absorption systems are, however, quite noisy on their own such that, from saturated region to saturated region, there is significant scatter about the mean-value curves shown in the top panel. In order to estimate how confidently we can distinguish the solid and dashed curves with a reasonable number of quasar spectra, we scale the number of identified absorption systems to what we would expect using 20 spectra. Specifically, we take the difference between the dashed and ∼ solid curves and divide by the scatter in each bin. The scatter of each bin is simply the

scatter in stacked transmission outside of large absorption systems, scaled by 1/ Nsat,large, added in quadrature with the scatter in the stacked transmission outside of smallp absorption

systems, scaled by 1/ Nsat,small. Here we scale to estimate the plausible scatter around the mean after estimatingp the transmission around saturated regions using 20 quasar absorption spectra. The results of this are shown in the bottom panel of Fig. 2.6 for the same ionization states. The results appear to be very encouraging, indicating that, assuming noiseless spectra, the solid and dashed curves are & 5σ statistically-significantly different (even for xHI = 0.05!). In addition, we see that the difference roughly follows a damping wing shape h i

109 2.7 Results

and remains significant for & 3000km/s. We should emphasize that, while the deuterium absorption feature will necessarily be a . 80km/s feature and require high resolution spectra to be seen, the damping wing feature extends an order of magnitude farther in velocity space and should be accessible to lower-resolution spectra. In Fig. 2.7, we show the same results as in Fig. 2.6, but assume a lower mean transmission of F = 0.05, consistent with spectra at 5.7 . z < 6 (Becker et al. 13). From the figure, h i we see that these results are very similar to those for F = 0.10, but with the significance h i curves peaking at a 70% lower value and with the stacked transmission recovering to a ∼ lower mean. Overall, this provides encouragement for applying the approach to higher-z spectra, suggesting that a range of physically interesting neutral fractions could be probed. It is also interesting to consider these results when spectra are generated according to the specifications of existing data. In Fig. 2.8 we show the same results as in the bottom panel of Fig. 2.6 except we have adjusted the spectra to mimic HIRES spectra. Namely, we have

assumed a spectral resolution with FWHM = 6.7km/s and bins with size ∆vbin = 2.1km/s (e.g. Viel et al. 178). Additionally, we have assumed a signal to noise of SNR = 10 at the continuum per 2.1km/s pixel and that we have 20 such spectra. While we are currently only aware of 10 such spectra, this case is still interesting since spectra with significantly worse spectral resolution should also be adequate for this test. From this figure, we can see that, despite the degradation of the spectra, the damping wing is still visible with the significance curve peaking at & 5σ (& 8σ) significance for the x = 0.22 (0.35) ionization state. However, this figure suggests that, in the x = 0.05 h HIi h HIi case, it is less-clear whether the damping wing is detectable. An important effect of adding noise to the mock spectra is that it obscures the precise location where spectra should be stacked and also increases the fraction of selected saturated regions which are, in fact, ionized. We find that for the spectra in this section 30%, 40%, ∼ and 75% of identified plausibly neutral regions are in in fact ionized for x = 0.35, 0.22, h HIi

110 2.7 Results

and 0.05, respectively. This is compared to 7%, 10%, and 20% contamination when noise ∼ is neglected. Statistical significances in this section are only estimates. In reality, the statistical significance with which the damping wing can be detected will depend on how extended the significance curves are, along with how correlated the errors in neighboring bins are. We discuss this in 2.8. § 2.7.2 Deuterium Feature Results

We now turn to consider the prospects for identifying deuterium absorption in realistic Ly β mock spectra. As discussed in 2.6, our aim is to identify plausibly neutral absorption § systems in the Ly β spectra and compare the stacked transmission moving blueward and redward away from the absorption. We identify the plausibly neutral regions in the same manner as for the damping wing. In Fig. 2.9, we show the results of applying the stacking approach to the same mock spectra as in the previous section. The top panel shows the mean transmission blueward (solid) and redward (dashed) moving away from plausibly neutral absorption systems for the same ionization states as in the previous section. We can very clearly see excess absorption in the partially neutral spectra, extending 80km/s, consistent with our expectations from ∼ Fig. 2.5. Additionally, we also find that, in the fully ionized case, the blueward and redward stacked transmission match up very well. As in the previous section, we can construct a rough significance curve for the difference between the blueward and redward transmission. Specifically, in the bottom panel of Fig. 2.9 we show the excess blueward absorption in units of the standard deviation of the stacked blueward transmission assuming 20 quasar spectra. We can see that the significance of the red/blue asymmetry extends 70km/s ( 0.3 Mpc/h) and is & 3σ for all of the ∼ ∼ partially neutral models considered, with increasing significance for models with higher neutral fractions. Additionally, we see that the curve corresponding to the fully ionized

111 2.7 Results

0.15

0.1

0.05 Stacked Transmission

0 0 1000 2000 3000 4000 5000 ∆ v (km/s) ) σ 20 = 0.35 HI = 0.22 15 HI = 0.05 HI 10 fully−ionized

5

0

Stacking Difference Significance ( 0 1000 2000 3000 4000 5000 ∆ v (km/s)

Figure 2.6: Ly α stacking results for various neutral fractions. The top panel shows the mean (noiseless) stacked transmission outside of large absorption systems (solid) and small absorption

systems (dashed) in the Ly α forest for neutral fractions xHI =0.35 (black), 0.22 (blue), 0.05 h i (red), and 0 (magenta). The transmission here is estimated from a large ensemble of mock spectra to obtain a smooth estimate of the average transmission around saturated regions in each model. The bottom panel shows the statistical significance of the difference between the dashed and solid curves in the top panel assuming a sample of 20 spectra are used in the stacking process. model shows no statistically-significant deviation from red/blue symmetry. Thus, this is indeed a very clean test for the presence of deuterium. However, the signal itself is an order of magnitude smaller in velocity-space extent and is found with significantly less statistical significance than the damping wing signal. Therefore, we expect that high-resolution, high- signal-to-noise spectra will be necessary to search for it.

112 2.7 Results

0.08

0.06

0.04

0.02 Stacked Transmission

0 0 1000 2000 3000 4000 5000 ∆ v (km/s) ) σ 15 = 0.35 HI = 0.22 HI 10 = 0.05 HI fully−ionized

5

0

Stacking Difference Significance ( 0 1000 2000 3000 4000 5000 ∆ v (km/s)

Figure 2.7: Ly α stacking results assuming F = 0.05. The above panels are identical to h i those in Fig. 2.6 except that mock spectra have been generated assuming F =0.05. h i

As in 2.7.1, we can reproduce Fig. 2.9 assuming spectra with specifications mimicking § Keck HIRES. Unfortunately, we find that, with a signal to noise per pixel in the continuum of 10, the deuterium feature is hard to observe. Because of this, we consider using 20 HIRES-style spectra with a signal to noise per pixel of 30 in the continuum. While this signal-to-noise value is higher than those for existing spectra we found in the literature, it is not unreasonable to assume such spectra may become available in the future. Furthermore, this may provide additional motivation to obtain such spectra. Regardless, after applying the stacking approach with twenty SNR = 30 HIRES spectra, we obtain the results shown in Fig. 2.10. This figure shows that the feature should be observable with modest statistical significance. Specifically, for x = 0.35 (0.22) the significance curve peaks at 3.7σ ( 3σ). HI ∼ ∼

113 2.7 Results ) σ 12 =0.35 HI =0.22 10 HI =0.05 8 HI =0 6 HI 4 2 0

Stacking Difference Significance ( 0 1000 2000 3000 4000 5000 ∆ v (km/s)

Figure 2.8: Results of Ly α stacking with HIRES-style spectra ( F =0.1). The above panel h i is identical to the bottom panel in Fig. 2.6 except that the spectra have had the bin size and spectral resolution adjusted to match that of Keck-HIRES spectra. Additionally, we have added noise such that the spectra have a signal-to-noise value of 10 per pixel at the continuum.

Additionally, when these curves are generated assuming MIKE-style spectra, with spectral

resolution FWHM = 13.6km/s and velocity bin size ∆vbin = 5.0km/s, we obtain similar curves as in Fig. 2.10 but with the signal being statistically significant over a smaller range of velocities. Again, important effects of adding noise to the mock spectra are that it obscures the precise location where spectra should be stacked and increases the fraction of selected plausibly neutral regions which are, in fact, ionized. We find that for the spectra in this section 15%, 20%, and 40% of identified plausibly neutral regions are in in fact ionized ∼ for x = 0.35, 0.22, and 0.05, respectively. This is compared to 7%, 10%, and 20% h HIi ∼ contamination when noise is neglected. As expected, we find a smaller level of contamination than in the previous section, owing to the increased signal to noise of the spectra used. However, for the case of deuterium, the effect of noise on the stacking location is more apparent. Fig. 2.9 demonstrates that, without noise, deuterium absorption imprints a feature on stacked noiseless spectra extending 80km/s, but only extending 60km/s in ≈ ≈ stacked noisy spectra, as shown in Fig. 2.10.

114 2.7 Results

The above results suggest that stacking Ly β transmission in high-z spectra can indeed be used to detect the presence of primordial deuterium, and hence that of hydrogen, but that high-resolution and high signal-to-noise spectra will be required. Nevertheless, it would certainly be interesting to apply this approach to existing HIRES/MIKE spectra as it provides an additional test, independent of the damping wing search, for the presence of underlying neutral hydrogen in the IGM. As such, a detection with modest levels of statistical significance could lend credence to a claimed detection of the HI damping wing.

2.7.3 Dark Gap Statistics

We now shift our focus away from stacking and toward the distribution in lengths of regions of saturated absorption (dark gaps). As discussed in 2.4, the dark gap size distribution § in quasar spectra should contain information about the underlying ionization state of the IGM. Specifically, in a more neutral IGM, the typical sizes of dark gaps should be larger and the shape of the dark gap size distribution should have a more gradual decline, and possibly show a hint of bimodality, toward large L. We continue this discussion in this section by considering plausible dark gap size dis- tributions that could be observed with moderate-resolution, moderate-signal-to-noise spec- tra. Specifically, we consider spectra with spectral resolution FWHM = 100km/s, bin size vbin = 50km/s, and a signal-to-noise ratio of 10 at the continuum. These spectra are of only slightly better quality than SDSS spectra. Additionally, since we are not concerned with Ly β transmission, we are able to use the entire Ly α forest for each spectra. In Fig. 2.11, we show the resulting dark gap size histogram expected for 20 such spectra for x = 0.35 (black), 0.22 (blue), 0.05 (cyan), and 0 (magenta). In generating this h HIi figure, we use the same ensemble of mock spectra as in 2.7.1 and 2.7.2, except with their § § spectral resolution and bin size modified as mentioned. We maintain the requirement that F = 0.1. h i

115 2.8 Forecasts

This figure qualitatively agrees with Fig. 2.3, where noiseless spectra with finer spectral resolution were used, but shows a shift toward larger L due to smoothing the spectra. Additionally, the ionization states are not as easily distinguishable as in Fig. 2.3, with the x = 0.05 distribution looking practically identical to the fully ionized scenario. h HIi However, for the other neutral fractions considered, the situation looks very encouraging. The distributions for x = 0.22 and 0.35 show a more gradual decline toward large L h HIi than the fully ionized case and also reveal the clear emergence of a bimodal distribution. Additionally, the largest dark gaps in these ionization states are roughly twice as large as in the fully ionized case.

2.8 Forecasts

Having discussed the results of the proposed stacking approaches applied to realistic mock spectra, we now consider the ability of these methods to constrain the ionization state of the z 5.5 IGM. Specifically, in this section we focus on the ability of each method to rule ∼ out the hypothesis of a fully ionized IGM. In both the case of the HI damping wing and deuterium absorption feature, we would like to compare models representing different ionization states and estimate the ∆χ2 between x = 0 models and the fully ionized model, assuming a reasonable number of spectra. h HIi 6 Let F (∆v) denote the mean behavior for a model with given neutral fraction, x , hxHIi h HIi as a function of stacked velocity separation and let Fion(∆v) denote the mean behavior of the ionized model, also as a function of stacked velocity separation. The precise definitions of what is meant by behavior will be discussed later in this section. In this case, the ∆χ2 value between two models can be calculated by

∆χ2 =∆F C−1 ∆F T (2.9) hxHIi hxHIi · · hxHIi where C is the covariance matrix of the x model, representing the correlation between h HIi T stacked pixels, and ∆Fhx i(∆v) Fion(∆v) Fhx i(∆v), with ∆F denoting its trans- HI ≡ − HI hxHIi

116 2.8 Forecasts

pose. For simplicity, rather than estimating the full covariance matrix and its inverse, we approximate pixels at sufficiently wide separations as independent. We then coarsely sample the pixels – on the scale where they can be well approximated as independent – and assume a diagonal covariance matrix for the coarsely sampled pixels. Specifically, we estimate ∆χ2 by simply adding up the squared statistical significance of each coarsely- hxHIi 2 sampled bin, ∆FhxHIi(∆vi)/σhxHIi(∆vi) , where σhxHIi(∆vi) is the standard deviation of

FhxHIi(∆vi).  

2.8.1 Deuterium

Perhaps it is best to consider the case of the deuterium absorption feature first. In the case of a fully ionized IGM, the transmission looking blueward and redward from absorption systems should be symmetric on average, with excess blueward absorption only occurring when the IGM is significantly neutral. Therefore, we may calculate the ∆χ2 between hxHIi

stacked transmission looking redward (FhxHIi,red(v)) and blueward (FhxHIi,blue(v)) from plau- sibly neutral absorption systems for each ionization state x to estimate our ability to h HIi rule out the hypothesis of a fully ionized IGM in each case. Thus, in the context of Eq. 2.9, we have

∆F (∆v) F (∆v) F (∆v) (2.10) hxHIi ≡ hxHIi,red − hxHIi,blue C−1 = δ /σ2 (v ) (2.11) ij ij hxHIi,blue i

where σhxHIi,blue(vi) is the standard deviation of the stacked transmission blueward of plau- sibly neutral absorption systems, assuming a given number of spectra, and we have assumed that we have already resampled ∆FhxHIi(v) at sufficient velocity separations such that neigh- boring bins can be approximated as independent. At this point, the only missing ingredient is the minimum separation between two stacked pixels for them to be considered indepen- dent. We calculate the correlation function between stacked pixels in Ly β, and find that the correlation has a width of FWHM 80km/s and, as such, we do not expect to get ≈

117 2.8 Forecasts

more than one independent bin within the scale of the deuterium feature. Therefore, a rough estimate of the ∆χ2 value obtainable in each ionization state can be estimated hxHIi simply by the peak value in the “significance curves” in Fig. 2.10. Thus, if the underlying neutral fraction of the IGM is x = 0.22 (0.35), then we expect to be able to rule out a h HIi fully ionized IGM with 3σ ( 3.7σ) confidence, assuming 20 HIRES/MIKE spectra with ∼ ∼ signal to noise of 30 per pixel at the continuum. Unfortunately, we do not expect to be able to rule out the hypothesis of a fully ionized IGM if the underlying neutral fraction is x . 0.05. h HIi

2.8.2 HI Damping Wing

Assessing the statistical significance with which we can observe the HI damping wing is slightly more complicated than the deuterium case since the test for its detection is not as simple. When faced with actual spectra, we would look for the presence of significant and extended absorption outside of large absorption systems compared to that outside of small absorption systems. Therefore, the behavior we would like to compare in each case is the stacked transmission

outside of plausibly neutral absorption systems (flarge(∆v)) and the transmission outside of

likely ionized absorption systems (fsmall(∆v)). Let us denote

F (∆v) f (∆v) f (∆v) (2.12) ≡ small − large

as the difference in these stacked transmissions where FhxHIi(∆v) and Fion(∆v) represent this behavior for the ionization state with neutral fraction x and the fully ionized state, h HIi respectively. Thus, in the context of Eq. 2.9, we have

∆F (∆v)= F (∆v) F (∆v) (2.13) hxHIi ion − hxHIi −1 2 C = δij/σF (vi) (2.14) ij hxHIi

118 2.9 Conclusion

2 where σF (vi) denotes the standard deviation of F (v) at vi. The resulting ∆χ hxHIi hxHIi value indicates the expected significance with which we could rule out a fully ionizedpIGM if the neutral fraction were, in fact, x . Again, for Eq. 2.14, we have assumed that h HIi

∆FhxHIi(∆v) has been resampled at velocity separations such that the pixels can be treated as independent. We find that the correlation function between pixels of stacked transmission in the Ly α forest within the scale of the HI damping wing has FWHM 100km/s. While ≈ this scale is large, the excess absorption due to the presence of damping wing absorption leaves a feature extending & 3000km/s, leaving us with & 30 independent bins within the scale of the feature. In this manner, we are able to calculate a rough estimate for the ∆χ2 values for the ionization states considered thus far. Assuming the same type of spectra as in Fig. 2.6, namely 20 HIRES spectra with signal to noise in the continuum of 10 per pixel, we find that if the IGM is, in fact, 5%, 22%, or 35% neutral, then we should be able to rule out a fully ionized IGM at 5.3σ, 19.2σ, or 26.3σ, respectively. In the case of F = 0.05, this h i reduces to 14.8σ, 8.7σ, and 2.2σ, respectively.1 While we are only aware of 10 such spectra ∼ that exist at the moment, we still regard this estimate as somewhat conservative. We found that excess stacked absorption due to the damping wing extends thousands of km/s, and therefore it is not necessary to have the state-of-the-art in spectral resolution to measure it. Especially with such extended correlation among neighboring pixels, it is unclear how much is gained by resolution improvements beyond 100km/s. ∼

2.9 Conclusion

In this work, we developed empirical tests of the possibility that the Epoch of Reionization is not yet complete by z 5.5. Specifically, we proposed three measurements that can ∼ be made with existing, and future, high-redshift quasar spectra in order to investigate this region of reionization history parameter space.

119 2.9 Conclusion

First, we discussed using the dark gap size distribution in quasar spectra as a means of constraining the z 5.5 neutral fraction. We find that not only do the typical sizes of dark ∼ gaps increase with neutral fraction but that the shape of the size distribution is also sensitive to the neutral fraction. Specifically, the presence of dark gaps sourced by significantly neutral hydrogen islands introduces a bimodality in the dark gap size distribution. We find that this bimodality should be observable at z 5.5, provided that x & 0.2, and should ∼ h HIi not be affected by continuum fitting errors. Next, we proposed a method for searching for hydrogen damping wing absorption by strategically stacking regions of transmission in the Ly α forest. Specifically, we searched for excess absorption in stacked transmission outside of plausibly neutral regions compared to that outside of likely ionized regions. We found that the presence of the hydrogen damping wing will result in excess absorption extending 5000km/s past ionization boundaries ∼ of neutral regions. Furthermore, this excess absorption should be detectable with & 5.3σ statistical significance for x & 0.05, using 20 HIRES-style spectra with a signal-to-noise h HIi value per pixel of 10 at the continuum. Lastly, we proposed a similar stacking measurement utilizing the Ly β forest in order to search for deuterium absorption associated with significantly neutral hydrogen islands at z 5.5. We proposed searching for this feature by looking for excess absorption in stacked ∼ Ly β transmission blueward of plausibly neutral regions compared to the corresponding redward transmission. We find that this feature should be observable in principle but will likely require additional high-resolution spectra in order to be detected. Specifically, we find that the feature should be observable at z 5.5 with 3σ ( 3.7σ) confidence using ∼ ∼ ∼ 20 HIRES-style spectra with a signal-to-noise value per pixel of 30 at the continuum if x = 0.22 (0.35). While we are not aware of this many available spectra with such h HIi specifications, this provides motivation for acquiring such spectra in the future, possibly through the follow-up observation of SDSS quasars.

120 2.9 Conclusion

While we have taken many steps to ensure that the analysis of mock spectra presented in this work is realistic, there are still additional complexities that will be faced when one is presented with actual spectra. For example, we treat all portions of our spectra as being at z = 5.5 when, in reality, the redshift will evolve along the lines of sight. In addition, we ignored spatial fluctuations in the UV radiation field and in the temperature density relation. Additional work will certainly be required to definitively interpret future measurements along the lines we suggest here. However, we believe the signatures explored here are well-worth further investigation and should ultimately improve our understanding of the reionization history of the IGM.

Appendix A: Contamination from DLAs?

A potential concern is that damping wings from super Lyman-limit systems and damped Ly- α absorbers (DLAs) might produce “false positives” and contaminate our search for diffuse neutral islands. Since DLAs and super Lyman-limit systems are mostly associated with galaxies and the circumgalactic medium (see Wolfe et al. 181 for a review), we would like to distinguish these absorbers from the more diffuse and spatially extended islands of neutral hydrogen that are the subject of our search. In addition, note that it is difficult to fully resolve and model high column density absorbers in cosmological simulations (e.g. Rahmati and Schaye 150 and references therein) – especially given our present aim of capturing the large-scale features of reionization – and so the impact of these systems is not captured in our present modelling. Fortunately, we don’t expect these dense absorbers to be a big contaminant, provided we make use of the Ly-β forest – which helps owing to the lower cross section in the wing of the line (compared to Ly-α) – and confine our search to fairly extended neutral islands. The Ly-β line profile for a high column density absorber can be approximated by a Lorentzian,

121 2.9 Conclusion

so that the optical depth at velocity offset ∆v is: σ R τ (∆v)= N β,0 β . (2.15) β,DLA HI π 2 2 (∆v/c) + Rβ

For comparison, a fully neutral and isolated absorber of co-moving length Lneut produces sat-

urated Ly-β absorption over a velocity extent slightly larger than ∆vneut = H(z)Lneut/(1 + z). We can then determine the column density required for a DLA to produce as long a

saturated region in the Ly-β forest as produced by a neutral island of length Lneut. We consider a DLA to produce saturated absorption at velocity separations where τ 3. β,DLA ≥ Provided the extent of the absorber is large enough that ∆v /c R (which is a good neut ≫ β approximation for the extended neutral islands of interest), this critical column density,

NHI,crit, is given by:

τ 1 + z L 2 N = 7.2 1021cm2 β,DLA neut . HI,crit × 3 6.5 3.8Mpc/h    h i (2.16)

The fiducial value of Lneut in the above equation corresponds to ∆vneut = 500 km/s – this is the minimum saturated stretch included in our stacks when we search for neutral regions (see 2.6). The column density N required for a DLA to produce this much saturated § HI,crit absorption is quite large, and the abundance of DLAs with column densities larger than

NHI,crit is very small. Quantitatively, taking the Gamma function fit to the column density distribution of DLAs from Prochaska et al. (148) 1 (which accounts for the sharp cutoff in the observed abundance of DLAs at high column densities), we find that the number of DLAs with N N is only dN(> N )/dz = 1.5 10−3. For reference, the redshift extent of HI ≥ HI,crit HI,crit × 1This is for the case where we do not attempt to further optimize the analysis for the decrease in transmission. It is possible that further gains could be made, with Fig. 2.7 representing a best-possible-case scenario. 1Specifically, we use their highest redshift bin fit, which includes DLAs between redshifts 3.5 ≤ z ≤ 5.5.

122 2.9 Conclusion

the forest between the Ly-α and Ly-β emission line at these redshifts is roughly ∆z 1, ≈ and so such high column density DLAs should be exceedingly rare. Since NHI,crit is only a little smaller than the exponential cut-off in the column density distribution function,

the results are rather sensitive to the precise choice of NHI,crit. Given that smaller column- density DLAs might still leak into our stack if they happen to be next to saturated ionized regions, it is worth checking this dependence. However, even choosing N = 2 1020 HI,crit × cm2 yields only dN/dz = 0.43, which is still smaller than the abundance of neutral islands we seek to detect. From these estimates, we expect very minimal contamination from DLAs leaking into our stacked sample of possible neutral regions. Note also that deuterium Ly-β absorption from these high column density absorbers will be in the saturated part of the HI Ly-β line, and so DLAs should not contaminate our search for the deuterium signature of neutral islands either. A separate possible worry is that DLAs could instead contaminate our sample of small saturated regions that we use for comparison purposes (as described in 2.6). Our small § saturated sample is meant to reflect absorption around saturated yet ionized regions, and so should not contain significant damping wing absorption. In principle, wings from any DLAs in this sample could influence the transmission around the small saturated regions. It seems unlikely that this is a significant worry, since the saturated yet ionized regions are likely vastly more abundant than even the small column density DLAs and super Lyman-limit systems. In addition, we can simply inspect the profile of the small saturated sample to see whether it shows any hint of damping wing absorption that might arise from either small isolated neutral regions or DLAs. Although contamination from DLAs does not appear to be a big worry for our tests, a more detailed examination would certainly be warranted if possible neutral islands are discovered in real data. We may also be able to remove DLA-contaminated regions by flagging spectral regions in the Ly-α and Ly-β forest that have the same redshift as strong metal absorbers, which generally accompany DLAs (see e.g. Wolfe et al. 181).

123 2.9 Conclusion

Appendix B: Further Utilizing the Ly β Forest

In order to infer the presence of the HI damping wing, we would like to compare the stacked transmission outside of plausibly neutral absorption systems to what that transmission would have been in the absence of the damping wing. Up to this point, we have been using the stacked transmission outside of small absorption systems as a proxy for the latter quantity. From there, we argued that any extended excess absorption outside of large, plausibly neutral absorption systems compared to small, likely ionized absorption systems is indicative of the presence of damping wing absorption. However, we do have another handle on what transmission would be like in the absence of the HI damping wing and that is the transmission in the Ly β forest. As discussed in 2.5.2, the HI damping wing should be significantly reduced in the Ly β forest compared to § the Ly α forest. Specifically, we saw that the damping wing optical depth in Ly β falls to less than one at velocity separations & 25km/s from neutral gas. Therefore, at separations greater than this, stacked transmission in the Ly β forest should provide information on what the shape of the Ly α transmission would have been in the absence of the damping wing, with foreground Ly α absorption only altering the stacked Ly β transmission by an overall constant F (z ) . Using information from stacked Ly β transmission has the appeal h α α i that it does not require using physically different regions of space in order to estimate the damping-wing-less transmission outside of selected plausibly neutral absorption systems. This provides protection from problems arising from unanticipated differences between the small, likely ionized absorption systems and the large, plausibly neutral absorption systems. Thus, we would like to find a way to estimate the Ly α transmission in the absence of the damping wing by using only the Ly β transmission. In principle, this can be done by using simulations to model the relationship between the two and generating a (ionization-state- dependent) mapping that takes a measurement of stacked Ly β transmission outside of large absorption systems and maps it to an estimate of the damping-wing-less Ly α transmission in the same regions. From there, the ratio of the stacked Ly α transmission to this estimate

124 2.9 Conclusion

of the damping-wing-less Ly α transmission would leave us with an estimate of e−τDW(v). In the left-hand panel of Fig. 2.12, we show the recovered e−τDW(v) curve after applying this approach to each of the ionization states considered thus far, and then normalizing each curve to peak at 1. Specifically, a mapping between stacked Ly β transmission and stacked damping-wing-less Ly α transmission for each ionization state was generated using a large ensemble of mock spectra and then applied to groups of 20 spectra. The error bars in the figure show the scatter in the estimated damping wing absorption between realizations of 20 spectra. For ease of viewing, we show only the error bars for x = 0 and 0.35. This h HIi figure demonstrates that the approach works well and recovers a damping wing shape for an IGM with x & 0.05 (in the absence of noise). h HIi A few things are worth pointing out about this process. First, the recovered damping wing profiles are only useful to the extent that they provide confidence that we are, in fact, observing neutral hydrogen in the IGM. The stacked profile of the HI damping wing is a complicated entity and, as such, we do not expect to be able to, for example, fit the recovered curves to Eq. 2.6 and estimate x . Secondly, it is comforting to note that h HIi not only is no damping shape recovered in the case of x = 0, but even if a mapping h HIi corresponding to a significantly neutral IGM is applied to a measurement of a fully ionized IGM, we do not recover a damping wing shape. Therefore, we do not expect this approach to yield false positives. Lastly, this process relies on simulations in order to map the stacked Ly β transmission to damping-wing-less Ly α transmission and is therefore somewhat model- dependent. However, we do not expect the specifics of reionization to significantly impact this mapping and we are also not interested in the fine details of the results here. We are primarily interested in whether damping wing absorption can be measured at all in the case of a somewhat neutral IGM and, as such, we are comfortable with this level of model dependence. Finally, we find that this mapping is relatively simple. Namely, for velocity separations & 100 200km/s, the stacked Ly β transmission and the stacked damping-wing-less Ly α −

125 2.9 Conclusion

transmission differ by roughly a constant multiplicative factor. Thus, in aiming to recover the shape of the stacked damping wing absorption, it appears to be a very good approx- imation to simply divide the stacked Ly α transmission by the stacked Ly β transmission. In the right-hand panel of Fig. 2.12 we simply take groups of 20 spectra and divide their stacked Ly α transmission by their stacked Ly β transmission (outside of large absorption systems) and give the result unity amplitude. Qualitatively, the results look very similar to those obtained from the mapping procedure (shown on the left-hand side) but are without any model-dependence. Additionally, we again find that, in the case of a fully ionized IGM, we do not recover a damping wing shape. Thus, this provides another check which may be performed with actual spectra in order to bolster confidence that a damping wing is in fact being observed. A potential concern for our Ly α stacking approach in general could be that, while we make the approximation that ionized regions are exposed to a uniform ionizing background, the ionizing background will in fact be fluctuating spatially. It is then possible that, in sce- narios where the ionizing background is weaker closer to the stacking location and stronger farther from the stacking location, an extended recovery could be imprinted on the stacked transmission despite the IGM being fully ionized. If one were not careful, and if these spatial fluctuations occurred on scales comparable to the damping wing feature, a false detection could be possible. One tool we have to protect against this is the fact that the scale of the damping wing is significantly smaller in Ly β than in Ly α. Therefore, any extended recov- ery in stacked transmission which occurs over similar scales in Ly α and Ly β is unlikely to be caused by the damping wing.

126 2.9 Conclusion

Appendix C: Extended damping wing Absorption from Cor- related HI Islands

As mentioned in 2.7.1, and seen in Fig. 2.6, 2.7, & 2.8, stacked Ly α transmission outside of § large saturated regions in a significantly neutral IGM displays excess absorption extending significantly past the scale of an isolated damping wing. To explain this, we seek to model the expected transmission outside of a neutral region, incorporating the correlation between the neutral region at the origin and neighboring neutral regions. In order to simplify the calculation, while capturing the main effect, we ignore correlations between the neighbors themselves – i.e., we only include the correlation between the neutral region at the origin and the neighbors and ignore inter-neighbor correlations. Ignoring correlations between the neighboring neutral regions, we can approximate them as following a Poisson distribution and consider the total absorption contributed by these neutral islands or “clouds” following Zuo and Phinney (198). Suppose that on average m clouds contribute to the absorption at a given region of the spectrum. Let F e−τ1 e−τ2 e−τk denote the transmission when k clouds reside along the line of sight k ≡ · · · and impact the given spectral region, with τi being the optical depth of the ith cloud. If we allow the clouds to be placed independently and if they have equal optical depths, then:

k F e−τ1 e−τ2 e−τk = e−τ1 e−τ2 e−τk = e−τ . (2.17) h ki≡ · · · · · ·  Using this expression, we can calculate the ensemble-averaged transmission by averaging over all the possible numbers of intervening clouds:

∞ ∞ ∞ e−m (me−τ )k F = Pois(k; m) F = mk e−τ k = e−m (2.18) h i h ki k! k! k=0 k=0 k=0 X −τ −Xτ X = e−meme = e−m(1−e ). (2.19)

127 2.9 Conclusion

Let us define the quantity τeff as

−τ e−m(1−e ) e−τeff (2.20) ≡ τ m 1 e−τ . (2.21) eff ≡ −  We now have the absorption from neighboring clouds, characterized by the parameter τeff. We just need to adapt this slightly to the problem at hand. Suppose that – with certainty – there is a neutral region located at v = 0 and let us consider the excess absorption, above random, contributed by neighboring neutral regions. First, the optical depth at v = ∆v that is contributed by a neighboring cloud located at v = v′ will depend on ∆v v′ : clouds | − | closer to v = ∆v will have larger optical depths at that corresponding frequency. We can account for this by substituting τ τ( ∆v v′ ), according to Eq. 2.6. Next, the expected → | − | number of HI islands in a region with velocity extent dv′ nearby our stacking location can be approximated as

m dv′ n (1 + ξ (v′)) (2.22) ≈ h HIi HI,HI where n is the average number of HI islands per interval dv′ and ξ (v′) is the cor- h HIi HI,HI relation function between the centers of neutral regions separated by v′. From here, we can model the effective optical depth at a given velocity separation due to neighboring HI islands, τeff(∆v), as

′ τ (∆v)= dv′ n 1+ ξ (v′) 1 e−τ(|∆v−v |) . (2.23) eff h HIi HI,HI − Z  h i In other words, the excess effective optical depth from neighboring systems involves the convolution of the absorption profile around each region with the correlation function of the neutral regions. This is analogous to the “two-halo” term in the halo model (e.g. Cooray and Sheth 38).

128 2.9 Conclusion

Thus, the model for the overall stacked transmission outside of neutral regions, also incorporating the damping wing from the central neutral region, becomes:

F (∆v)= e−τDW(∆v)e−τeff(∆v). (2.24) h i This model requires two inputs. First, it requires the correlation function between the ′ centers of neutral regions, ξHI,HI(v ), which can be calculated from a model of the underlying ionization field. Second, the optical depth profile for neutral regions, τ( ∆v v′ ), is a | − | function of the size of the neutral regions, per Eq. 2.6. While the neutral regions in the mock spectra take on a range of sizes, we make the simplifying approximation here – but not in the body of this chapter – that the neutral regions at a given neutral fraction have one typical size and denote this Ltypical with a corresponding extent in velocity space vext. This effectively results in the optical depth profile of an individual neutral island in the right hand side of Eq. 2.23 being described by a piecewise function

if ∆v < v /2 ∞ | | ext τDW(∆v)= τ R c 1 1 (2.25)  GP α otherwise.  π ∆v vext/2 − ∆v + vext/2  −  Next, we would like to compare this model against results using mock spectra. We do this by first generating mock spectra which only include absorption from neutral islands, since this is the only type of absorption incorporated in our model, and stack these mock spectra at the HI/HII boundaries. To be clear, while the model curve described above adopted a

fixed Lneut for the purpose of calculating a τeff, the mock spectra here are generated using the same simulated ionization fields as throughout the rest of the chapter, with a wide range of sizes for the underlying neutral regions. To obtain a model for the stacked transmission, we first calculate the correlation function between the centers of neutral regions using the mock underlying ionization fields and also

choose a value for Ltypical to be used in the optical depth profile. Additionally, in Eq. ′ 2.23, the (1 + ξHI,HI(v )) term effectively breaks our integral into two pieces: the first

129 2.9 Conclusion

representing the mean absorption from neutral regions and the second representing the excess or reduced absorption at v = ∆v due to the clustering of neutral islands. For our case, we are only concerned with the excess absorption, so we make the replacement (1 + ξ (v′)) ξ (v′). HI,HI → HI,HI ′ Therefore, by providing a value for Ltypical and measuring ξHI,HI(v ), we can get a es- timate for the mean transmission outside of neutral regions which incorporates absorption from spatially-correlated neighboring regions. In the left panel of Fig. 2.13, we plot an example of this for x = 0.22. We show the modelled damping wing absorption from h HIi the central neutral region in blue, the modelled absorption from neighboring neutral islands and their damping wings in cyan, and the product of these in black. For comparison, we show the stacked transmission in the mock spectra in magenta, shifted by vext/2 to account for stacking occurring at HI/HII boundaries instead of at the center of neutral regions. All curves have been divided by the mean transmission. After taking Ltypical = 3.2 Mpc/h, we find good agreement between the above model and the stacked transmission. The precise agreement should be taken with a grain of salt, since the model makes several simplify- ing assumptions, especially that the neutral regions have a fixed size. However, the model does demonstrate that clustered neutral islands should lead to extended excess absorption, significantly beyond the scale of an individual damping wing. In the right panel of Fig. 2.13, we show the comparison between the stacked transmission (solid) and modelled transmission (dashed) for x = 0.35 (black), 0.22 (blue), and 0.05 h HIi (cyan), where each curve has been multiplied by the mean transmission for clarity. In generating these plots, we have assumed Ltypical = 2.5 Mpc/h, 3.2 Mpc/h, and 0.75 Mpc/h for x = 0.35, 0.22, and 0.05, respectively. We again find a very nice agreement between h HIi the modelled and stacked transmission, further confirming that spatially-correlated regions are indeed responsible for the significantly-extended excess absorption.

130 2.9 Conclusion

0.1

0.05 Stacked Transmission

0 0 20 40 60 80 100 v (km/s)

8 = 0.35 HI ) σ = 0.22 6 HI = 0.05 HI 4 fully−ionized

2

Statistical Significance ( 0

0 20 40 60 80 100 ∆ v (km/s)

Figure 2.9: Deuterium Ly β stacking results for various neutral fractions. The top panel shows the mean ensemble-averaged noiseless stacked transmission moving blueward (solid) and redward (dashed) away from large absorption systems in the Ly β forest for neutral fractions

xHI =0.35 (black), 0.22 (blue), 0.05 (cyan), and 0 (magenta). The bottom panel shows the ex- h i cess blueward absorption in units of the standard deviation of the stacked redward transmission, assuming 20 spectra.

131 2.9 Conclusion

4

) = 0.35

σ HI = 0.22 3 HI = 0.05 HI 2 fully−ionized

1

Statistical Significance ( 0

0 20 40 60 80 100 ∆ v (km/s)

Figure 2.10: Results of Ly β stacking with HIRES-style spectra. The above panel is the same as in the bottom panel of Fig. 2.9, except that it is generated using HIRES-style spectra, with spectral resolution of FWHM = 6.7km/s and additive noise with signal to noise of 30 per 2.1 km/s pixel at the continuum.

1 10

x = 0.35 HI x = 0.22 HI Scaled Frequency x = 0.05 HI x = 0 HI

0 10 2 3 4 10 10 10 L (km/s)

Figure 2.11: Mock dark gap size distribution. This figure is identical to Fig. 2.3 except that it uses spectra with spectral resolution FWHM = 100km/s, bin size ∆vbin = 50km/s, and a signal-to-noise ratio of 10 at the continuum. This figure shows the expected histogram of dark gap sizes using 20 spectra with xHI =0.35 (black), 0.22 (blue), 0.05 (cyan), and 0 (magenta) h i at fixed F =0.1. h i

132 2.9 Conclusion

1.5 x = 0 x = 0.05 x = 0.2 x = 0.35 HI HI HI HI DW τ 1 −

0.5 Estimated e

0 0 200 400 600 800 1000 ∆ v (km/s)

1.2 x = 0 x = 0.05 x = 0.2 x = 0.35 HI HI HI HI 1 DW τ 0.8 −

0.6

0.4 Estimated e 0.2

0 0 200 400 600 800 1000 ∆ v (km/s)

Figure 2.12: Using the Ly β forest to estimate damping-wing-less Ly α transmission. The above figure shows the estimated shape of stacked damping wing absorption for xHI = 0 h i (magenta), 0.05 (cyan), 0.22 (blue), and 0.35 (black). The curves have been normalized to have their mean values peak at 1. Additionally, we show error bars for the fully ionized case and

xHI = 0.35 case which indicate the scatter in the curves between groups of 20 spectra. The h i top plot is obtained by using a large ensemble of mock spectra to model a mapping between stacked Ly β transmission and stacked damping-wing-less Ly α transmission and then applying this to groups of 20 spectra. Meanwhile, the bottom figure plots the ratio of the stacked Ly α flux to the stacked Ly β flux, providing a simplified estimate of the damping wing contribution to the absorption for each case.

133 2.9 Conclusion

= 0.35 1 HI 1 = 0.22 HI 0.8 = 0.05 0.8 HI

0.6

0.6 τ − e / τ

− 0.4 e 0.4

Stacked Transmission 0.2 1−Halo 0.2 2−Halo 1−Halo + 2−Halo 0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 ∆ v (km/s) ∆ v (km/s)

Figure 2.13: Model for the extended damping wing absorption. The left panel shows the components of our model for stacked transmission outside of a neutral region compared to the stacked transmission using mocked spectra (magenta) for xHI = 0.22. We show the h i absorption due to the central neutral region (blue), average absorption due to neighboring, clustered neutral regions (cyan), and the product of the two transmissions (black). These are denoted in the legend as “1-Halo”, “2-Halo”, and “1-Halo + 2-Halo” in analogy with the halo model. In the right-hand panel, we show the comparison between the modelled transmission

(dashed) and transmission from stacked mocked spectra (solid) for xHI = 0.35 (black), 0.22 h i (blue), and 0.05 (cyan). The curves in the right-hand figure have been multiplied by the mean transmission (computed here ignoring resonant absorption for illustration). In this appendix, the stacking is done at the HI/HII boundaries and only damping wing absorption is incorporated to demonstrate the extended excess absorption owing to correlated neighboring systems.

134 Chapter 3

Preliminary Stacking Results

In this chapter we briefly present preliminary results obtained from applying the stacking methods described in 2 to spectra provided to us by Andrei Mesinger, Ian McGreer, and § Valentina D’Odorico. These spectra are described in McGreer et al. (104),1 with the basic properties shown in their Table 1. We have recreated their table here (Table 3.1) for convenience. When working with actual spectra, there are a few details that we must deal with which we did not need to discuss while using mock spectra. Specifically, the actual spectra have varying spectral resolutions, varying signal-to-noise values, do not have periodic boundary conditions, and are not at a fixed redshift. We deal with the differing spectra resolutions by smoothing all spectra with a Gaussian kernel with full width at half max equal to 50 km/s and resampling them at this resolution. We deal with the varying signal-to-noise values by incorporating an inverse-variance weighting scheme when averaging transmission in different regions of different spectra. Typically, when implementing an inverse-weighting scheme, the “variance” is the noise variance. However, in our case we have two effective sources of noise: noise in the spectra themselves and resonant absorption throughout the

1Technically, we use all of the spectra in McGreer et al. (104) except for J0002+2550 and MMT obser- vations of J1137+3549. This detail is reflected in Table 3.1.

135 Table 3.1: Overview of quasar spectra used in our preliminary stacking tests.

Object z z t (hr) τ α source AB exp h eff,limi J1420-1602 5.73 19.7 4.00 5.3 MagE J0927+2001 5.77 19.9 0.33 3.8 ESI J1044-0125 5.78 19.2 4.79 5.2 MagE J0836+0054 5.81 18.7 0.33 4.7 ESI 4.00 3.9 MMT 2.27 5.9 XShooter J0840+5624 5.84 19.8 0.33 4.1 ESI J1335+3533 5.90 20.1 0.33 3.8 ESI J1411+1217 5.90 19.6 1.00 3.7 ESI J0148+0600 5.92 19.4 10.00 6.3 XShooter J0841+2905 5.98 19.8 0.33 3.5 ESI J1306+0356 6.02 19.5 0.25 4.3 ESI 11.50 5.4 XShooter J0818+1722 6.02 19.6 4.50 4.6 MMT 5.90 5.7 XShooter J1137+3549 6.03 19.5 0.67 3.8 ESI J2054-0005 6.04 20.7 11.00 4.4 MagE J0353+0104 6.05 20.5 1.00 3.5 ESI J1630+4012 6.07 20.4 4.39 3.2 MMT J0842+1218 6.08 19.6 0.67 4.0 ESI J1509-1749 6.12 20.3 6.00 4.7 MagE 8.32 5.2 XShooter J1319+0950 6.13 20.0 10.00 5.7 XShooter J1623+3112 6.25 20.1 1.00 4.2 ESI J1030+0524 6.31 20.0 10.32 5.3 ESI 7.46 5.4 XShooter J1148+5251 6.42 20.1 11.00 6.0 ESI

Note: τ α is the median effective optical depth in the Ly α forest for a pixel (binned to h eff,limi 3.3 cMpc) with a flux equivalent to the 1σ noise estimate.

136 spectra. In other words, if we are trying to measure an underlying damping wing signal, then additional resonant absorption occurring in the span of the damping wing is effectively a source of noise for us. To incorporate this, we perform the inverse-variance weighting using 2 a variance σtot defined by

σ2 σ2 + σ2 (3.1) tot ≡ N F

2 2 where σN is the spectrum’s noise variance and σF is the variance in the flux of the spectra due to actual absorption, calculated after smoothing all spectra to a common resolution and after binning in redshift. Thus, each region of transmission incorporated into the stack is 2 weighted by the 1/σtot value associated with that spectrum. To accommodate the fact that the spectra evolve in redshift along the line of sight, we perform stacking in two discrete redshift bins, one incorporating 5.5 z 5.7 and one ≤ gap ≤ with 5.7 z 6, where z is the redshift of the dark gap that we are stacking at ≤ gap ≤ gap the boundaries of. To incorporate the fact that the spectra do not have periodic boundary conditions, we require that regions of transmission extend for at least ∆vmin = 5000km/s before terminating at the end of the spectrum. This is to prevent somewhat artificial noise in the stacked transmission resulting from regions of transmission that encountered the edge of the spectral coverage before spanning the entire velocity range of the stack. With the exceptions of these caveats, though, we perform the stacking in the same qualitative manner as described thus far. Namely, when searching for the HI damping wing, we stack Ly α transmission outside of dark gaps in the Ly β forest and set the minimum length of a “large” dark gap to be Lmin = 300km/s and the maximum length of a “small” dark gap to be Lmax = 300km/s in Ly β. These precise values differ somewhat from those used earlier in order to increase statistics. Based on the signal-to-noise values, number, and spectral resolution of the spectra in Table 3.1, we do not expect to be able to detect

137 absorption due to deuterium. However, we perform the search anyway and stack Ly β transmission outside of any dark gaps in Ly β whose length are at least Lmin = 100km/s. In Fig. 3.1 and Fig. 3.2, we show the preliminary Ly α stacking results for the low-z and high-z bins, respectively. In each figure, the top panel shows the results of stacking outside of small dark gaps while the bottom panel shows the results of stacking outside of large dark gaps. In each case, the solid lines are for mock spectra generated to roughly mimic the spectral resolution and signal-to-noise characteristics of the true spectra.1 The colors match those in earlier plots, namely, x = 0 (magenta), 0.05 (cyan), 0.22 (blue), and 0.35 h HIi (black). The stacked transmission for the spectra in Table 3.1 is shown in dashed green. There are two comparisons that we can make here. First, we can compare the stacked transmission outside of large and small dark gaps to the simulated stacked transmission under different neutral fractions. For this comparison, let us first focus on stacked trans- mission outside of small dark gaps. For both redshift bins, the stacked appears relatively noisy and not well-fit by any of the simulated curves. While the stacked transmission in this case appears larger for ∆v > 3000km/s than for ∆v < 3000km/s, the behavior is not consistent with our expectations for that from damping-wing absorption. Specifically, we would expect the smallest stacked transmission to occur for very small velocity separations, as can be seen in the simulated stacked transmission outside of the large dark gaps. How- ever, in dashed-green curve, there is a spike in stacked transmission for velocity separations of less than a few hundred km/s. It is unclear if this excess absorption is a genuine feature or simply the result of statistical fluctuations. One possibility is that, if absorption sys- tems are clustered, the excess absorption could be akin to the “2-halo” term described in Appendix C.

1The model spectra used here actually result in a mean flux that is smaller than we get when stacking spectra from Table 3.1. We apply a scaling factor to the mock spectra’s stacked flux in order for the two to match. We have generated mock spectra with a range of hF i . 0.1 and find that this approximation is appropriate.

138 97 Gaps Contributed =0 0.30 =0.05 =0.22 0.25 0.35

0.20 Real Spectra

0.15

0.10 Stacked Transmission Stacked

0.05

0.00 0 1000 2000 3000 4000 5000 v (km/s)

58 Gaps Contributed =0

0.20 =0.05 =0.22 0.35 0.15 Real Spectra

0.10 Stacked Transmission Stacked 0.05

0.00 0 1000 2000 3000 4000 5000 v (km/s)

Figure 3.1: The above figure shows the results of stacking Ly α transmission outside of dark gaps in the Ly β portion of the spectrum with L< 300km/s (top) and L> 300km/s (bottom)

for dark gaps with 5.5 zgap 5.7. The solid curves are generated using mock spectra assuming ≤ ≤ xHI = 0 (magenta), 0.05 (cyan), 0.22 (blue), and 0.35 (black). The dashed green line shows h i the stacking results for the spectra described in Table 3.1.

The stacked transmission outside of large dark gaps in both the low-z and high-z bins appear less noisy. In the lower redshift bin, the stacked transmission is relatively flat but starts off large and falls toward the mean transmission by ∆v 3000km/s. This behavior ≈ is not precisely the same as for the mock spectra but does not show any excess absorption

139 225 Gaps Contributed 0.20 =0 =0.05

0.15 =0.22 0.35 Real Spectra

0.10

Stacked Transmission Stacked 0.05

0.00 0 1000 2000 3000 4000 5000 v (km/s)

138 Gaps Contributed 0.20 =0 =0.05

0.15 =0.22 0.35 Real Spectra

0.10

Stacked Transmission Stacked 0.05

0.00 0 1000 2000 3000 4000 5000 v (km/s)

Figure 3.2: This figure is identical to Fig. 3.1 except we stack outside of dark gaps with

5.7 zgap 6. ≤ ≤

that can be attributed to the damping wing. Meanwhile, in the high-z bin, the stacked transmission is much flatter. While the stacked transmission shows a slight increase over the course of ∆v 500 5000km/s, the transmission is initially rather high – not suggestive ∼ − of damping-wing absorption. If the stacked transmission from the mock spectra is reliable, this curve is consistent with a neutral fraction of x . 0.05. However, since the analogous h HIi curve for the low-z bin did not match well with the model curves, this comparison is likely risky.

140 These comparisons have the disadvantage that they rely on simulations and are therefore somewhat model-dependent. Earlier in this chapter we proposed directly comparing the stacked transmission outside of large and small dark gaps as a test for the damping wing. If damping-wing absorption was significant, we would expect the associated absorbers to result in large dark gaps, such that stacked transmission outside of large dark gaps would show significant excess absorption to that outside of small dark gaps. Performing this comparison by eye for the top/bottom panels of Fig. 3.1 and Fig. 3.2, we see the opposite behavior: stacked transmission outside of small dark gaps shows more absorption than that outside of large dark gaps. While we do not have an explanation for this behavior, it is not suggestive of a damping wing. Lastly, in Fig. 3.4 and Fig. 3.3 we compare the stacked Ly β transmission redward (blue) and blueward (green) of dark gaps in the Ly β forest with Lgap > 100km/s. We should emphasize that, due to the number and quality of the spectra used, we do not expect to observe excess absorption due to deuterium, even for a significantly-neutral IGM. However, we perform the test anyway and find that the stacked redward and blueward transmission agree very well. Therefore, while we have not performed a rigorous statistical analysis of the stacked transmission using these spectra, after a preliminary look we do not see any obvious evidence of neutral hydrogen at z < 6. With further work, this analysis should yield an interesting upper limit on the fraction of the IGM volume in the neutral phase.

141 Figure 3.3: This figure shows the results of stacking Ly β transmission outside of dark gaps with L > 100km/s in the spectra described in Table 3.1. For this figure, we stack outside of dark gaps with 5.5 zgap 5.7. ≤ ≤

Figure 3.4: This figure is identical to Fig. 3.3 except we stack outside of dark gaps with

5.7 zgap 6. ≤ ≤

142 Chapter 4

On Modelling and Measuring the Temperature of the z 5 IGM ∼

4.1 Introduction

The temperature of the low density intergalactic medium (IGM) after reionization retains information about when and how the gas was heated during the Epoch of Reionization (EoR) (e.g. Hui and Gnedin 73, Hui and Haiman 74, Miralda-Escud´eand Rees 124, Theuns et al. 171). The temperature of the IGM in turn impacts the statistical properties of the Ly-α forest towards background quasars and so the absorption in the forest provides “fossil” evidence regarding the timing and nature of reionization. Scrutinized carefully, this fossil may therefore improve our understanding of reionization. For example, the IGM will likely be cooler at z 5 if most of the IGM volume reionized at relatively high redshift, near e.g. ∼ z 10, than if reionization happened later, near say z 6. If reionization occurs early, ∼ ∼ the gas has longer to cool and reaches a lower temperature than if it happens late, at least provided the gas is heated to a fixed temperature at reionization. In addition, the IGM temperature should be inhomogeneous, partly as a result of spatial variations in the timing of reionization across the universe (Cen et al. 30, Furlanetto and Oh 58, Trac et al. 177).

143 4.1 Introduction

Careful measurements of the IGM temperature after reionization should hence constrain the average reionization history of the universe, and may potentially reveal spatial variations around the average history as well. Two separate phases of reionization are likely relevant for understanding the thermal history of the IGM: an early period of hydrogen reionization during which hydrogen is ionized, and helium is singly ionized by star-forming galaxies, and a later period in which helium is doubly-ionized by quasars, i.e. HeII reionization. Hydrogen reionization completed sometime before z 6 or so (e.g. Fan et al. 52, although it might conceivably end as late ∼ as z 5 – see Lidz et al. 88, McGreer et al. 105, Mesinger 116) , while mounting evidence ∼ suggests HeII reionization finished by z & 2.5 3 (see e.g. Syphers et al. 169, Worseck et al. − 182 and references therein). Many of the existing IGM temperature measurements focus on redshifts of z 2 4 (Lidz et al. 86, McDonald et al. 102, Ricotti et al. 153, Schaye et al. ∼ − 160, Theuns et al. 173, Zaldarriaga et al. 195); in this case the temperature is likely strongly influenced by HeII reionization (e.g. McQuinn et al. 111) and so these measurements mostly constrain helium reionization rather than hydrogen reionization. In order to best constrain hydrogen reionization using the thermal history of the IGM, temperature measurements at higher redshift (z & 5) are required. Indeed, recent work has started to probe the temperature at these early times. In particular, the recent study by Becker et al. (10) includes a measurement at z = 4.8; Bolton et al. (17) and Raskutti et al. (151) determine the z 6 temperature in the special “proximity zone” region of the ∼ Ly-α forest close to the quasar itself; and the analysis in Viel et al. (178) starts to bound the z & 5 IGM temperature, although these authors focus on placing limits on warm dark matter models. The temperature at these higher redshifts is unlikely to be significantly impacted by HeII reionization. In addition, the “memory” of intergalactic gas to heating during the EoR gradually fades and so measurements as close as possible to the EoR should, in principle, be most constraining. It is not, however, obvious that the IGM temperature can be measured

144 4.1 Introduction

accurately enough from the z & 5 Ly-α forest to exploit the sensitivity of the high redshift temperature to the properties of reionization. In particular, the forest is highly absorbed by z 5 with z & 6 spectra showing essentially complete Gunn-Peterson (Gunn and Peterson ∼ 65) absorption troughs (Becker et al. 13, Fan et al. 52). An interesting question is then: what is the highest redshift at which it is feasible to measure the IGM temperature from the Ly-α forest? Towards this end, the goal of this chapter is to both model the thermal state of the z 5 IGM, incorporating inhomogeneities in the hydrogen reionization process, and to ∼ quantify the prospects for actually measuring the IGM temperature using z & 5 Ly-α forest absorption spectra. The outline of this chapter is as follows. In 4.2, we describe the numer- § ical simulations used in our analysis. In 4.3, we present plausible example models for the § reionization history of the universe and describe our approach for modeling inhomogeneous reionization. We adopt a semi-analytic approach for modeling the resulting thermal history of the IGM, as described in 4.4. In this section, we also quantify the statistical properties § of the temperature field in several simulated reionization models. Finally, in 4.5 we discuss § how to measure the temperature from the z 5 Ly-α forest, and forecast how well it may ∼ be measured with existing data. Our main conclusions are described in 4.6. § This work partly overlaps with previous work which also recognized the importance of, and modeled, temperature inhomogeneities in the z 5 IGM and considered some of the ∼ observable implications (Cen et al. 30, Furlanetto and Oh 58, Trac et al. 177).1 One key difference with this earlier work is that we consider a more direct approach for measuring the temperature of the z 5 IGM from the Ly-α forest. Our modeling of the thermal state ∼ of the IGM is closely related to that in Furlanetto and Oh (58), except that we implement a similar general approach using numerical simulations, which allow us to construct mock Ly-α forest spectra and to measure the detailed statistical properties of these spectra.

1Lai et al. (82) also considered temperature fluctuations from hydrogen reionization, but these authors focused on z ∼ 3 where – as they discussed – these fluctuations should be small and swamped by effects from HeII reionization.

145 4.2 Simulations

The works of (Cen et al. 30, Trac et al. 177) use radiative transfer simulations to model hydrogen reionization and the thermal history of the IGM and so these authors track some of the underlying physics in more detail than we do here. However, our approach here is faster, simpler, and more flexible, while we believe that it nevertheless captures many of the important processes involved.

4.2 Simulations

Our analysis makes use of two different types of numerical simulations. First, we use the “semi-numeric” scheme of Zahn et al. (189) to model reionization; this algorithm is per- formed on top of the dark matter simulation of McQuinn et al. (108). The McQuinn et al. (108) simulation tracks 10243 dark matter particles in a simulation volume with a co-moving sidelength of 130 Mpc/h. Using the semi-numeric technique allows us to capture the impact of inhomogeneities in the reionization process, while providing the flexibility to explore a range of possible reionization models. In these models, we assume that the gas distribution closely traces the simulated dark matter distribution. We discuss the impact of this approx- imation when relevant. As we will describe, we map between the redshift of reionization of each gas element and its temperature at high redshift using the technique of Hui and Gnedin (73); this mapping depends on the density of each gas element. We then produce mock Ly-α forest spectra, according to the usual “fluctuating Gunn-Peterson approxima- tion” (e.g. Croft et al. 39, Miralda-Escude et al. 123) although here we additionally account for the temperature variations from inhomogeneous reionization. We also make use of one of the smoothed hydrodynamic (SPH) simulations from Lidz et al. (86). These simulations were run using the code Gadget-2 (Springel 168). This simulation tracks 2 10243 particles (with equal numbers of dark matter and baryonic × particles) in a 50 Mpc/h simulation box. In these calculations, we ignore the inhomogeneity of the reionization process. We use these simulations to more faithfully capture the gas

146 4.3 Reionization Histories

distribution (for gas elements that reionize at a given time). In constructing mock Ly- α forest spectra from these simulations, we first modify the simulated gas temperatures, according to various prescriptions, in order to test how sensitive the statistical properties of the absorption are to the gas temperature.

4.3 Reionization Histories

In an effort to explore how the thermal state of the post-reionization IGM depends on the reionization history of the universe, we consider several example reionization histories. Our aim is to consider models that result in a wide range of possible thermal histories, while broadly maintaining consistency with current observational constraints on reionization. For simplicity, we assume (as is common) that early galaxy populations produce ionizing photons at a rate that is directly proportional to the rate at which matter collapses into halos above some minimum mass. The minimum mass describes the host halo mass above which 9 galaxies form readily; here we adopt Mmin = 10 M⊙. We compute the collapse fraction from the Sheth-Tormen halo mass function (Sheth et al. 166). With these assumptions, the volume averaged ionization fraction ( x ) obeys the following differential equation (Madau h ii et al. 95, Shapiro and Giroux 165):

d x df x h ii = ζ coll h ii. (4.1) dt dt − t¯rec The first term on the right hand side of the equation describes the rate at which neutral atoms are ionized, while the second term on the right hand side accounts for ionized atoms that recombine. The recombination time (t¯rec) depends on the clumpiness of the IGM, parametrized by a “clumping factor”, C = n2 / n 2, and the temperature of the IGM. h ei h ei In solving this equation – and for this purpose only – we assume an isothermal IGM. We approximate the clumping factor and the temperature as independent of redshift. Adopting the case-B recombination rate in solving this equation, a temperature of T = 2 104 K, and 0 ×

147 4.3 Reionization Histories

C = 3 (see e.g. McQuinn et al. 112, Pawlik et al. 139 for a discussion regarding plausible values of the clumping factor) gives

3 1+ z −3 T 0.7 t¯ = 0.93Gyr 0 . (4.2) rec C 7 2 104K     ×  Solving the differential equation, Eq. 4.1, suffices to compute the average ionization fraction as a function of redshift, given the minimum mass and efficiency, ζ, of the ionizing sources. In order to model reionization inhomogeneities, we use the “semi-numeric” scheme of Zahn et al. (189), which is based on the excursion set model of reionization developed in Furlanetto et al. (62). This scheme captures the tendency for halos – and hence galaxies – to form first in regions that are overdense on large scales, and to reionize before more typical locations in the universe. In the simplest version of the semi-numeric scheme, recombina- tions are considered only in an average sense and are treated as simply reducing the overall efficiency at which atoms are ionized. Let us denote the resulting efficiency factor as ζ˜(z) to distinguish it from the above ionizing efficiency factor ζ. As we explain subsequently, we allow this efficiency factor to be redshift dependent. We can then consider the condition that a region of co-moving size R is ionized. In the initial conditions, the mass enclosed within this co-moving region is M = 4πR3 ρ /3, with ρ denoting the average matter h M i h M i density per co-moving volume. The condition for this region to be ionized is then:

ζ˜(z)f (> M δ , M) 1. (4.3) coll min| M ≥ In this equation f (> M δ , M) is the conditional collapse fraction, i.e., the fraction coll min| M of mass in halos above the minimum mass (Mmin) in a region of linear overdensity δM . Here

δM denotes the overdensity when the linear density field is smoothed on mass scale M. In order to tabulate a reionization redshift for many grid cells across the volume of our simulation, we smooth the density field – linearly evolved from the initial conditions – on a range of mass scales, starting from large scales and stepping downward until we reach the scale of each simulation cell. For each simulation cell, and across all smoothing

148 4.3 Reionization Histories

scales considered, we record the highest redshift at which the ionization barrier (Eq. 4.3) is crossed. This highest crossing redshift is considered to be the reionization redshift, zr, for the cell in question. We tabulate reionization redshifts for each of 5123 grid cells. This provides us with a reasonable model for the expected spatial variations in the redshift of reionization – and the coherence scale of these inhomogeneities – across the simulation volume. Note that here we approximate the excursion set model as determining the redshift at which each volume element is reionized, although in reality mass elements move from their initial positions, and overdense regions expand less rapidly than typical locations. This approximation is commonly made, and is reasonable given the large size of the ionized regions (Furlanetto et al. 62) and the correspondingly large coherence scale of the spatial variations in the reionization redshift. Another ingredient we use from the McQuinn et al. (108) simulation is the evolved non- linear dark matter density field, interpolated onto the same grid (using CIC interpolation) as the reionization redshifts. For our calculations with this simulation, we generally assume that the gas distribution follows the simulated, gridded dark matter distribution. Note that the smoothing introduced by gridding the dark matter particles is comparable to the Jeans 4 smoothing scale: the co-moving Jeans wavenumber for isothermal gas at 10 K, is kJ = 13h −1 Mpc at z = 5 which is comparable to the Nyquist wavenumber of the grid, kNyq = 12h Mpc−1. More relevant, however, is the “filtering scale” – essentially a time-averaged Jeans scale – and this should be smaller than the Jeans scale by around a factor of a few (Gnedin and Hui 64). In any case a single global smoothing only roughly approximates the full effect of Jeans smoothing. We will return to discuss this further in 4.5 and 4.5.3. In particular, in § § order to approximately capture the impact of small scale structure and thermal broadening in our mock quasar spectra, we will add small-scale structure using a lognormal model. Although using the gridded dark matter density field to represent the gas distribution is

149 4.3 Reionization Histories

inadequate for making detailed mock spectra, it suffices for our model of the temperature distribution of the low density gas. Returning to further consider the semi-numeric modeling, an important caveat is that this algorithm does not return precisely the expected volume-averaged ionization fraction (see the Appendix of Zahn et al. 189 for a discussion). Here we simply tune ζ˜(z) to produce the desired redshift evolution of the ionization fraction. Although this procedure is not ideal, small adjustments to the ionizing efficiency factor have little impact on the size of the ionized regions at a given volume-averaged ionization fraction, x , and so this approach is h ii adequate for our present purposes. The redshift evolution of the volume-averaged ionization fractions are shown in Fig. 4.1 for three example models. The symbols show the average ionized fraction from the simulation cube at different redshifts. We call the three examples in the figure the “Low-z” model, the “Mid-z” model, and the “High-z” model. The Mid-z model adopts a redshift dependent efficiency factor of the form ζ˜(z) = 35(1 + z/13)1.75 for z 12 and ζ˜ = 35 for ≤ z > 12. For comparison, the black dashed line shows the solution to Eq. 4.1 for a model with 9 ζ = 46, C = 3, and Mmin = 10 M⊙. Hence the semi-numeric scheme in the Mid-z model has been tuned to return the ionized fraction expected from Eq. 4.1 for a plausible model. The Low-z and High-z models are similar to the Mid-z model, except that the efficiency factor in the semi-numeric models has been adjusted to ζ˜(z) = 12(1 + z/11)0.60 at z 10 and to ≤ ζ˜(z) = 12 at z > 10 for the Low-z model and to a constant efficiency factor ζ˜(z) = 70 for the High-z model. (Although these alternative models were not themselves explicitly tuned to match particular solutions to Eq. 4.1, the general behavior is similar to in the Mid-z model except that reionization happens a little later/earlier in the Low-z/High-z model and so these models also appear reasonable). It is also useful to quantify the timing and duration of reionization, as well as the optical depth to Thomson scattering (τe), in each model. Defining the “completion” of reionization as the redshift where the volume averaged ionization fraction first reaches x = 1, the h ii

150 4.3 Reionization Histories

Figure 4.1: Example reionization histories. The red triangles show the simulated volume- average ionization fraction in our semi-numeric High-z reionization model, the black squares are for the Mid-z reionization scenario, and the blue pentagons are for a low redshift (Low-z) reionization model. The black dashed line shows the reionization history computed by solving 9 Eq. 4.1 with ζ = 46, Mmin = 10 M⊙ and C = 3. The semi-numeric efficiency parameters ζ˜(z) in the Mid-z case have been tuned to match this model.

151 4.3 Reionization Histories

High-z model completes at z = 9.6, the Mid-z model at z = 6.7, and the Low-z model at z = 5.8. As one measure of the “duration” of reionization, we consider the redshift spread over which x evolves from x = 0.1 to x = 1. This duration is ∆z = 2.7, 4.3, 3.5 for h ii h ii h ii the High-z, Mid-z, and Low-z models. Note that the duration is the longest in the Mid-z model because the ionizing efficiency factor ζ˜(z) has the strongest redshift dependence in

this case. The electron scattering optical depths are τe = 0.088, 0.066, 0.052 for the High-z, Mid-z, and Low-z reionization models respectively. These values assume that the fraction of helium that is singly ionized is identical to the fraction of hydrogen that is ionized, and ignore the slight boost expected from the free electrons produced after HeII reionization, which we do not track in this work.

The present constraint on τe from Planck CMB temperature anisotropy data (Ade et al. 2), combined with the E-mode polarization power spectrum at large angular scales from +0.012 WMAP nine year data (Bennett et al. 14), is τe = 0.089−0.014. Most of the constraining power here comes from the WMAP polarization data. Hence our High-z model produces a τ close to the presently preferred value, the Mid-z model is low by 1.6 σ, while the e − Low-z model is too low by 2.6 σ. Hence our lower redshift reionization models are already − marginally disfavored, but they are still certainly worthy of further investigation. The Planck collaboration should soon announce new large-scale CMB polarization measure- ments; the improved frequency coverage of the Planck satellite should help guard against

foreground contamination, and further test these models for τe. Although the current τe constraints allow higher reionization redshift models than the three examples considered here, the z 5 IGM temperature is insensitive to the reionization redshift if reionization ∼ happens above z & 10 ( 4.4, Hui and Haiman 74). While viable, we need not consider such § models explicitly here since in these cases the z 5 temperature will be similar to that in ∼ our High-z model.

152 4.4 The Thermal State of the IGM

4.4 The Thermal State of the IGM

We now explore how the thermal state of the z 4 6 IGM depends on the reionization ∼ − history of the Universe, using the example histories of the previous section. In this section, we focus mostly on the Low-z and High-z models since they span a fairly wide range of possibilities for the IGM temperature at the redshifts of interest. The key equation describing the thermal evolution of a gas element in the IGM is (e.g. Hui and Gnedin 73): dT 2T dδ T dµ = 2HT + + dt − 3(1 + δ) dt µ dt 2µm + p (H Λ) . (4.4) 3ρkB − The first term on the right hand side accounts for adiabatic cooling owing to the over- all expansion of the universe. The second term describes adiabatic heating/cooling from structure formation, i.e. from gas elements contracting/expanding. In the third and fourth terms, µ is the mean mass per free particle in the gas, in units of the proton mass. The third term accounts for the temperature change that occurs because the mean mass per particle may change with time. H describes the heating term, while Λ is the cooling function of the gas. These terms describe the heat gain and loss per unit volume, per unit time, in the gas. Let us first summarize the qualitative behavior of the solutions to Eq. 4.4, focusing on the low density intergalactic gas that fills most of the volume of the universe (see also Furlanetto and Oh 58, Hui and Gnedin 73, Hui and Haiman 74, Miralda-Escud´eand Rees 124). During reionization, most gas elements are rapidly ionized and change their ionization fraction by order unity.1 The excess energy of the ionizing photons (above the ionization threshold) goes into the kinetic energy of the outgoing electrons, which quickly share their

1Sufficiently overdense regions/clumps may be only gradually ionized as reionization proceeds and the ionizing radiation field incident upon them grows in intensity, but we will neglect these, assuming that partly neutral clumps fill only a small fraction of the volume within mostly ionized regions.

153 4.4 The Thermal State of the IGM

energy with the surrounding gas, and raise its temperature. The first thing to consider is hence the initial temperature reached at reionization. Provided the gas becomes highly ionized, its temperature boost during reionization depends only on the shape of the spectrum – and not the amplitude – of the radiation that ionized it. In detail, we expect gas elements to be ionized by radiation with a range of spectral shapes. This should be the case both because the intrinsic ionizing spectrum will vary from galaxy to galaxy, and because the spectral shape may be hardened by intervening absorption, which will itself vary spatially depending on the column density of neutral gas between an ionizing source and an absorber. On the other hand, the ionized regions during hydrogen reionization likely grow under the collective influence of numerous (yet individually faint) dwarf galaxies (e.g. Robertson et al. 156), and so some of these variations may average down, provided gas elements are ionized by a combination of several sources and the ionizing radiation arrives along various different pathways. In any case, modeling the precise temperature input during reionization and its spatial variations requires full radiative transfer simulations and is well beyond the scope of our approach here. We adopt this uniform temperature boost approximation throughout, and discuss plausible values for the input temperature subsequently. Note also that in this case the temperature boost during reionization is independent of density: extra heat is put into the overdense regions since more atoms are ionized in these regions, but the heat is shared across the larger number of particles in the overdense parcel. After a gas element is reionized, it settles into ionization equilibrium and the UV ra- diation from the ionizing sources keeps the gas highly ionized (at least for the low density gas parcels that fill most of the volume of the IGM). In ionization equilibrium, each re- combination is balanced by a photoionization and the ionizations in turn heat the gas; the average time between recombinations in the low density IGM is long, and so the heat input from photoionization is significantly reduced after a parcel becomes highly ionized during reionization. In addition, the spectral shape of the ionizing radiation incident on a typical

154 4.4 The Thermal State of the IGM

gas element should soften – i.e., the average heat input per photoionization should drop – after reionization as the optical depth to ionizing photons decreases (Abel and Haehnelt 1).1 The dominant cooling processes are adiabatic cooling from the expansion of low den- sity gas parcels and Compton cooling off of the CMB. As a result of cooling, although gas elements that reionize at the same time start off with identical temperatures, irrespec- tive of their density, parcels with differing densities will not stay at the same temperature. In particular, the low density elements expand and cool faster than typical regions, while overdense regions recombine faster and thus – in ionization equilibrium – gain more heat from photoionizations after reionization. In addition, sufficiently overdense regions will be heated by adiabatic contraction. Hui and Gnedin (73) showed that this competition between adiabatic cooling/heating, Compton cooling, and photoionization heating, drives the intergalactic gas to generally land on a tight temperature-density relation of the form γ−1 T = T0(1 + δ) . Both the temperature at mean density, T0, and the slope of the tem-

perature density relation, γ, depend on the reionization redshift; T0 falls off and γ becomes steeper as the gas cools after reionization, until the gas gradually loses memory of the heat- ing during reionization. In the previous work of Hui and Gnedin (73), however, all of the gas was assumed to reionize at the same redshift. Here we would like to generalize this to incorporate spatial variations in the redshift of reionization (see also Furlanetto and Oh 58 and Hui and Haiman 74).

4.4.1 Modeling the Thermal State

In general, to follow the thermal evolution in Eq. 4.4 we should combine this equation with equations specifying the evolution of the ionized fraction of each different particle species.

1In reality, the spectral softening depends on how progressed reionization is globally since the hardening from absorption depends on the density and ionization state of all of the gas between a source and an absorber. Here we neglect this by fixing the spectral shape incident on each gas element after it is ionized.

155 4.4 The Thermal State of the IGM

However, for our present application a simpler approach should suffice. In particular, we start by assuming that each parcel is heated to a common temperature, Tr, at its reionization redshift, zr. We then follow the subsequent thermal evolution after a gas element is ionized by assuming ionization equilibrium and that each element is highly ionized (as in Furlanetto and Oh 58). More specifically, we assume that both HI and HeI are highly ionized, but that HeII is not yet ionized, i.e., that HeII reionization starts later than the high redshifts of interest for our study. We further assume the gas is composed of only hydrogen and helium, neglecting metal line cooling, and also molecular hydrogen cooling, which should be very good approximations for the low density IGM. In addition to adiabatic heating/cooling and Compton cooling, we track HI photoheating, HeI photoheating, and recombination cooling of HII/HeII using the rate expressions in the Appendix of Hui and Gnedin (73). We ignore collisional ionizations, and HI/HeI/HeII line excitation cooling: neglecting these processes should be a good approximation for the low density and highly ionized gas that fills most of the IGM volume after reionization. Furthermore, we ignore other potential heating sources such as shock heating, galactic winds, blazar heating, etc..(see e.g. Chang et al. 31, Hui and Haiman 74 and references therein for a discussion). In the Appendix we also derive approximate solutions using linear perturbation theory (incorporating only HI photoheating, Compton cooling, and adiabatic cooling/heating), that are useful for fast and fairly accurate estimates (see also Hui and Gnedin 73). In principle we could calculate the adiabatic expansion/contraction term (second term in Eq. 4.4) directly from the McQuinn et al. (108) simulation, at least under the approximation that the gas distribution traces the simulated dark matter density field. Here we instead follow the approach of Hui and Gnedin (73) and compute this term for tracer elements assuming their density evolution obeys the Zel’dovich approximation (Zel’dovich 197). As mentioned earlier, we do, however, extend this calculation to consider gas elements with a range of different reionization redshifts. The basic premise here is that gas elements with identical reionization redshifts should

156 4.4 The Thermal State of the IGM

land on a well-defined temperature-density relation (as supported by the tests in Hui and Gnedin 73 and subsequent work); we can determine this relation by solving Eq. 4.4 for many sample gas parcels. Incorporating, however, the spread in reionization redshifts, and that the reionization redshift of each parcel may correlate with its density, a perfect temperature-density relation will not generally be a good description. In other words, we follow sample gas parcels to determine the mapping between the temperature-density rela- tion at a given redshift and the reionization redshift and temperature, i.e., this is used to determine T (z z , T ) and γ(z z , T ). These mappings can then be applied to our reion- 0 | r r | r r ization simulation to determine the temperature of any gas element, given its reionization redshift and overdensity. To determine T (z z , T ) and γ(z z , T ), we follow the thermal 0 | r r | r r evolution for 20, 000 tracer elements for many different reionization redshifts, assuming their density evolves according to the Zel’dovich approximation, and fit separate power-laws for each z, zr, and Tr. In the Zel’dovich approximation, the density field evolves according to the equation: 1 1+ δ = , (4.5) det[δij + D(t)ψij] with ψij denoting the initial deformation tensor, and D(t) denoting the linear growth factor (normalized to unity today). The density evolution of a tracer element can then be specified by the eigenvalues of the local initial deformation tensor. As in Hui and Gnedin (73) and Reisenegger and Miralda-Escude (152), we can construct realizations of the density evolution in the Zel’dovich approximation by randomly drawing eigenvalues of ψij from the expected probability distribution (Doroshkevich 47). We do this following Bertschinger and Jain (16), Hui et al. (75). In our fiducial model, we take the temperature at reionization to be T = 2 104 K (see r × Furlanetto and Oh 58, McQuinn 106 for a discussion of this choice). In calculating the pho- toionization heating term after a gas parcel reionizes, we assume that the specific intensity of the ionizing radiation is a power-law in frequency close to the hydrogen photoionization

157 4.4 The Thermal State of the IGM

threshold, J(ν) ν−α with α = 1.5. As mentioned previously, the heat input is insensitive ∝ to the amplitude of the ionizing radiation, provided the gas is highly ionized. This spectral shape is intended to be somewhat harder than expected for the intrinsic spectrum of the ionizing sources, since intervening absorption will harden this spectrum (Furlanetto and Oh 58, Hui and Haiman 74, Zuo and Phinney 199).

4.4.2 Simulated Temperature Field

We now examine the properties of the simulated temperature field, modeled as described in the previous section. First, we consider the mapping between the temperature at mean density, T0, and the slope of the temperature-density relation, γ, for gas at various redshifts, given the reionization redshift, zr, of each gas element. This is shown, for our baseline set of assumptions, in Fig. 4.2 for each of z = 4.5, 5.0, and z = 5.5. The values of T0 are close to the temperature at reionization (T = 2 104 K) for gas elements that ionized at redshifts r × just above z = 5.5, since these elements have had very little time to cool. On the other hand,

gas parcels with higher zr have had longer to cool and are hence at lower temperatures.

For instance, gas elements that reionized at zr = 8 have cooled down to T0 = 8, 800 K by z = 5.5, more than a factor of two below the temperature at reionization. The temperatures of gas elements that reionize at sufficiently high redshift, however, become insensitive to

the precise redshift of reionization. In particular, gas elements that reionize above zr & 10

are all at T0 = 6, 700 K at z = 5.5, irrespective of zr. This results mainly because Compton cooling is very efficient at high redshift (z & 10), and effectively erases the memory of the photoheating during reionization (Hui and Gnedin 73). Indeed, this is the main reason that we don’t consider still higher redshift reionization models, although they would be allowed by the present τ constraints as discussed in 4.3 (but perhaps disfavored by other data e § sets, see e.g. Kuhlen and Faucher-Giguere 80, Robertson et al. 156 for recent summaries.): the thermal state of the IGM is insensitive to higher redshift reionization models.

158 4.4 The Thermal State of the IGM

Figure 4.2: Thermal state of gas elements with a given reionization redshift, as a function of that redshift. In each case, the gas elements are heated to a temperature of T = 2 104 r × K during reionization, and the residual photo-heating after reionization is computed assuming that the (hardened) spectral index of the ionizing sources is α =1.5 near the HI photoionization edge. Top panel: The temperature at mean density (T0) for gas elements at each of z =4.5, 5.0 and 5.5 as a function of their reionization redshift. Bottom panel: This is similar to the top panel, except it shows the slope of the temperature-density relation (γ 1) rather than T0. − Note that although we assume that gas elements with a given reionization redshift all land on a well defined temperature-density relation, this will not generally be a good description once we account for the spread in reionization redshift across the universe.

159 4.4 The Thermal State of the IGM

12.8 16563

12.1 15363

11.5 14164

10.8 12953

10.1 11754

9.47 10555

8.8 9344

8.13 8145

7.46 6934

6.79 5735

6.12 4535

Figure 4.3: Reionization redshifts and temperatures at z = 5.5 in the low-z reionization model. Left panel: The reionization redshifts for a narrow slice (0.25 Mpc/h thick) through the simulation. Each slice is 130 Mpc/h on a side. The red regions indicate locations with the highest reionization redshifts across the simulation slice, while the dark regions are the last to be reionized. Right panel: The temperature of the same slice as in the top panel. The red areas in this panel show the hottest locations in the slice, and correspond to the dark regions in the top panel that are reionized late. The dark blue regions in the temperature slice, on the other hand, are the coolest regions that reionized first. The color scales are chosen so that 99% of simulation cells in the slice shown here have redshifts and temperatures falling between the minimum and maximum values on the color bar.

The bottom panel is similar to the top panel except here we plot γ 1 versus z . Gas − r elements that reionize just above z = 5.5 are close to isothermal, while elements that ionize at z & 10 have a steeper slope, γ = 1.53. The T and γ 1 curves at z = 4.5 and z = 5.0 r 0 − illustrate less sensitivity to zr, since gas elements at these redshifts have had longer to cool down from their initial temperatures at reionization. Nevertheless, the models at these lower redshifts still certainly do show some dependence on zr. We then use the curves plotted in Fig. 4.2 as a mapping to predict the temperature of various grid cells in our simulation given their overdensities, δ, and reionization redshifts, zr. This procedure allows us to model the temperature field across the entire simulation volume at various redshifts. Figs. 4.3 and 4.4 show the result of applying the mapping (at z = 5.5) to the simulated

160 4.4 The Thermal State of the IGM

15.4 16468

14.8 15166

14.3 13863

13.7 12548

13.2 11245

12.7 9943

12.1 8627

11.6 7325

11 6010

10.5 4707

9.97 3405

Figure 4.4: Reionization redshifts and temperatures at z = 5.5 in the high-z reionization model. Identical to Fig. 4.3, except this figure shows the contrasting High-z model. Note that the color scale in this case also encompasses 99% of the reionization redshifts and temperatures in the simulation slice, but that these ranges are different than in the previous figure.

density field.1 Specifically, these figures show thin slices (0.25 Mpc/h thick) through the simulation volume, with the top panel showing the reionization redshift and the bottom panel the corresponding temperature of cells in the simulation volume. In the Low-z model (Fig. 4.3), the temperature field has sizable spatial variations on large scales. As anticipated earlier, these result because of the spread in the timing of reionization across the universe. As one can infer from the slice, the regions that are at low-density (when the density field is smoothed on large-scales) – i.e., the “voids” in the density distribution – are the last to reionize. These regions are at the highest temperature shortly after reionization because they have had the least amount of time to cool (see also Furlanetto and Oh 58, Trac et al.

1In practice, we apply the mapping to the simulated density field at slightly higher redshift (z = 6.9) since we don’t currently have outputs from this simulation at the lower redshifts of interest. Using the higher redshift output artificially reduces the variance in the density field, and the resulting structure in the temperature field somewhat. For our present purposes, this is not important. The main effect of boosting the density variance should be to increase the minimum and maximum density contrasts shown in scatter plots such as Fig. 4.5. Importantly, this has little impact on the median temperature-density relation and the scatter around this relation for the range of density contrasts in our scatter plots. We have tested this explicitly using a lognormal approximation to the density field at z = 4.5 and z = 5.5.

161 4.4 The Thermal State of the IGM

177). In contrast, the temperature field in the High-z model (Fig. 4.4) has mostly lost memory of the heating during reionization and so the temperature variations are more subtle here. This is expected from Figs. 4.1 and 4.2: much of the gas in this model is reionized at zr & 10, and efficient Compton cooling mostly removes the memory of reionization in this case. The temperature variations that are apparent in the High-z model instead reflect the usual temperature-density relation, as the competition between cooling and heating after reionization drives overdense regions to larger temperatures. These temperature variations are primarily coherent on the Jeans/filtering scale and so, as evident from the simulation slices, these fluctuations are concentrated mostly on smaller scales than the ones induced by the spread in the timing of reionization. A further, more quantitative description is provided by constructing scatter plots of the temperatures of many simulated gas elements as a function of their densities. This is shown in Figs. 4.5 and 4.6 for the Low-z and High-z reionization models, respectively. Broadly similar results may be found in earlier work by Trac et al. (177) and Furlanetto and Oh (58). The red short-dashed line in each figure shows the median gas temperature at z = 5.5, while the green long-dashed line is the median temperature at z = 4.5. Fig. 4.5 shows that the temperature of the z = 5.5 IGM is generally rather high – and has a large amount of scatter at low densities – in the Low-z reionization model. By contrast, the temperature in the High-z reionization model (Fig. 4.6) is smaller – e.g., by 60% for the median temperature near the cosmic mean density – as is the scatter. In the Low-z model the median temperature is a fairly flat function of density for gas less dense than the cosmic mean. Note that although the regions that have low density – when the density field is averaged on large scales – ionize last and are mostly hotter than denser regions (see Fig. 4.3), this does not fully “invert” the temperature-density relation. This is because the density field on the scale of the simulation grid (and at the Jeans scale) is only somewhat correlated with

162 4.4 The Thermal State of the IGM

Figure 4.5: Temperature density relations at z = 4.5 and z = 5.5 in the Low-z reionization model. The blue points show the temperature and density of gas elements from the simulation at z =5.5, while the black points are the same at z =4.5. The red short dashed line shows the median simulated temperature as a function of density at z =5.5. The green long dashed line is the same at z =4.5.

163 4.4 The Thermal State of the IGM

Figure 4.6: Temperature density relations at z = 4.5 and z = 5.5 in the High-z reionization model. Identical to Fig. 4.5, except the results here are for the High-z reionization model.

164 4.4 The Thermal State of the IGM

the larger scale density variations that determine the spread in the timing of reionization in our model. In any case, in agreement with previous work (Furlanetto and Oh 58, Trac et al. 177), the usual temperature-density relation is a poor description of the thermal state of the IGM in the Low-z model at z = 5.5. At slightly lower redshifts, z = 4.5, the temperature has dropped somewhat and the scatter in the Low-z model has partially subsided, although it is still substantial. The median temperatures in the two reionization models are closer to each other by z = 4.5, but they still differ by 30%. It is also useful to calculate the power spectrum of temperature fluctuations in each model. Since we are assigning a value of T0 and γ to each grid cell in the simulation

volume, we can easily consider the power spectrum of T0 rather than the power spectrum of the full temperature field. The advantage of considering the power spectrum of T0 is that this power spectrum vanishes in the case of homogeneous reionization. In the case of homogeneous reionization, the temperature is a power-law in the gas density, and so the full temperature field still has (mostly small scale) fluctuations sourced directly by density

inhomogeneities. Hence we consider here the power spectrum of T0(x), or more precisely the power spectrum of δ (x) = (T (x) T )/ T . Although the power spectrum of this field T0 0 − h 0i h 0i is not directly observable, it nevertheless helps to characterize the temperature fluctuations from reionization. The power spectra in some of our models are shown in Fig. 4.7. Specifically, the curves show ∆2 = k3P (k)/(2π2), the contribution to the variance of δ per natural logarithmic T0 T0 T0 interval in k, i.e., per dln(k). In the Low-z model, the temperature fluctuations peak at a level of around ∆2 (k) 15%. We should keep in mind, however, that the scatter in the T0 ≈ temperature atq lower density is larger than at mean density (see Fig. 4.5). As a result, the power spectra of T0 shown here hence do not fully capture the impact of inhomogeneous reionization, but they do nevertheless illustrate the spatial scale of the reionization induced inhomogeneities as well as their redshift and model dependence. The power spectra (∆2 ) T0 are evidently fairly flat functions of k. This is not surprising, since the power spectra of

165 4.4 The Thermal State of the IGM

Figure 4.7: Power spectrum of temperature fluctuations in various models. The curves show x x the power spectrum of δ 0 ( ) = (T0( ) T0 )/ T0 from the simulated models. The blue T − h i h i dotted line, the black solid line, and the red short-dashed line are the power spectra at z =5.5 in the Low-z, Mid-z, and High-z models respectively. The black long-dashed line shows the δT0 power spectrum at z = 4.5 in the Mid-z model to illustrate how the temperature fluctuations fade with time.

166 4.4 The Thermal State of the IGM

the fluctuations in the ionization field are also rather flat functions of k during most of the

EoR (e.g. McQuinn et al. 110). At the same redshift, the δT0 power spectrum in the Mid-z Model is 3 times smaller in amplitude than in the Low-z model, while the amplitude of ≈ variations (∆2 (k)) in the High-z model are 300 times smaller than in the Low-z model. T0 ≈ As discussed previously, the small fluctuations in the High-z model result because Compton cooling is efficient at high redshift and this rapid cooling effectively erases the memory of heating at higher redshifts. Comparing the black solid and dashed lines illustrate how the fluctuations fade from z = 5.5 to z = 4.5 in the Mid-z model. These models illustrate the dependence of the thermal history of the IGM on the timing of reionization; let us briefly summarize our main findings here. The IGM temperature for models in which a significant fraction of the IGM volume is reionized at relatively low redshift, near z 6, is correspondingly larger than if most of the gas is reionized ∼ at higher redshift. In addition, the late reionization models produce sizable temperature inhomogeneities with fluctuations on scales as large as tens of co-moving Mpc. ∼

4.4.3 Variations around Fiducial Parameters

Before we proceed to discuss the observable signatures of the IGM temperature models, it is interesting to consider how variations around our fiducial assumptions regarding the reionization temperature, Tr, and the shape of the ionizing spectrum after reionization might impact the resulting thermal state of the IGM. To investigate this, we consider models where the reionization temperature is T = 3 104 K and T = 1 104 K to contrast with our r × r × fiducial model in which T = 2 104 K. The low T model requires sources with extremely r × r soft ionizing spectra, and is meant to represent a lower limit to the plausible reionization temperature, while the higher temperature T = 3 104 K case is more reasonable (e.g. r × McQuinn 106). In addition, for our fiducial reionization temperature we produce models with (post reionization) spectral shapes of α = 0.5 and α = 2.5 to compare with our baseline assumption of α = 1.5.

167 4.4 The Thermal State of the IGM

First, we consider how our models for T (z = 5.5 z ) and γ(z = 5.5 z ) depend on the 0 | r | r reionization temperature and spectral shape, i.e., we regenerate the models of Fig. 4.2 for different values of Tr and α. The results of these calculations are shown in Fig. 4.8. The

first feature to note is that T0 and γ are independent of zr and Tr in the limit of large reionization redshift: efficient cooling wipes out the memory of the early heating history. On the other hand, the z = 5.5 temperature is naturally quite sensitive to the reionization temperature if reionization occurred relatively recently. One consequence of this is that the scatter in the z 5 temperature will be larger in low redshift reionization models ∼ for cases with larger reionization temperatures: a high reionization temperature increases the temperature contrast between recently reionized gas parcels and those that reionized early. The increased scatter in these models may potentially boost the observability of the temperature inhomogeneities induced by spatial variations in the timing of reionization, as we explore subsequently.

Another important point is that increasing Tr in a high reionization redshift model will not help to mimic the z 5 temperature in a lower redshift reionization model, since the gas ∼ that reionized at high redshift reaches an asymptotic temperature that is insensitive to Tr. 4 On the other hand, decreasing Tr (as in the Tr = 10 K curves) in a low reionization redshift model will certainly diminish the distinction between this model and higher reionization redshift models. As we will see, however, the larger scatter and flatter trend of temperature

with density in the low zr, low Tr model offer potential handles for distinguishing between these models and higher reionization redshift scenarios. Next we consider how the results vary with changes in the spectral shape, α, as shown in the figure. These variations have a relatively minor effect. Adopting a harder ionizing spec- trum (smaller α) after reionization increases the amount of residual late-time photoheating. This thereby raises the asymptotic temperature and the asymptotic value is reached earlier. Quantitatively, the asymptotic temperature is 20% higher in the α = 0.5 case than in our fiducial α = 1.5 model, and 18% smaller for α = 2.5. The dependence on α is relatively

168 4.5 Measuring the Temperature of the z 5 IGM ∼ mild compared to other uncertainties in our modeling and so we don’t consider it further here. Another perspective is to construct scatter plots in the temperature-density plane and plot median temperature-density relations for various models, as in Fig. 4.5 and Fig. 4.6. This is shown for gas at z = 5.5 in Fig. 4.9. Consider first the two High-z models (the bottom two sets of points and dashed lines in the figure), which show the results of assuming T = 2 104 K (bottom-most case with red points and a black dashed line), and T = 3 104 r × r × K (shown as blue points and a cyan line, just above the bottom-most model). This shows that the results in this model are insensitive to Tr: this is as anticipated from Fig. 4.8. The next model is a Low-z case and has T = 1 104 K (green points and blue line). r × This model is clearly closer to the High-z reionization model than our fiducial Low-z case, which is shown in the figure as the upper most black points with red line fit. However, the median temperature at low density and the scatter in the temperature are both larger in the Low-z, low reionization temperature model than in the High-z models. If the scatter in the temperature, and the trend of temperature with density can be measured observationally, this may help break the partial degeneracies between reionization redshift and temperature.

4.5 Measuring the Temperature of the z 5 IGM ∼ We now turn to consider the impact of the thermal state of the IGM on the properties of the z 5 Ly-α forest, and on the possibility of extracting these signatures to learn ∼ about reionization. The effects of temperature on the statistics of the Ly-α forest are dis- cussed, for example, in Lidz et al. (86). The three main effects are: higher temperatures produce more Doppler broadening; the recombination rate of the absorbing gas is temper- ature dependent with hotter gas recombining more slowly, leading to less neutral gas and less absorption; hotter gas leads to more Jeans smoothing, with the precise impact of this smoothing depending on the entire prior thermal history of the absorbing gas (Gnedin and Hui 64).

169 4.5 Measuring the Temperature of the z 5 IGM ∼

The enhanced Doppler broadening and Jeans smoothing in models with high tempera- tures each act to reduce the amount of small-scale structure in the Ly-α forest. These two effects are not, however, entirely degenerate: Jeans smoothing filters the gas distribution in three dimensions, while Doppler broadening smooths the optical depth field along the line of sight (e.g. Zaldarriaga et al. 195). Previous studies suggest that Doppler broadening impacts the small scale structure in the forest more than Jeans smoothing, at least near z 3 (Lidz et al. 86, Peeples et al. 140, Zaldarriaga et al. 195). At z 5, we expect Jeans ∼ ∼ smoothing to have more impact, however: the high opacity in the Ly-α line at these redshifts implies that even slight density enhancements can give rise to noticeable absorption lines, and these slight density variations may be erased by Jeans smoothing. Unfortunately, it is challenging to model the impact of Jeans smoothing while incorporating a realistic model for inhomogeneous reionization and photoheating. This requires hydrodynamic models that resolve the filtering scale, while capturing a large enough volume to model patchy reion- ization. Furthermore, the filtering scale depends on the entire prior thermal history. In this chapter, we defer this challenge to future work and assume that the effect of Jeans smoothing is sub-dominant to that of Doppler broadening. We caution that Jeans smooth- ing might, however, enhance the impact of patchy reionization and modeling it may be necessary to robustly interpret future measurements. In any case, the small-scale structure in the Ly-α forest should be sensitive to the thermal state and the thermal history of the IGM, and so we now consider an approach for estimating the amplitude of small-scale structure in the forest. Here we will use the basic technique described in Lidz et al. (86), except applied here to simulated data at higher redshift where there is more absorption in the forest. The first issue we aim to explore here is to what extent the temperature of the IGM is measurable at higher redshift, where the forest is significantly more absorbed. A second goal is to explore the impact of the temperature inhomogeneities modeled in the previous section. We briefly outline the approach of Lidz et al. (86) for measuring the small-scale structure

170 4.5 Measuring the Temperature of the z 5 IGM ∼

– and thereby extracting constraints on the IGM temperature – here for completeness. In this approach, each spectrum is convolved with a Morlet wavelet filter and the smoothing scale of this filter is tuned to extract the amplitude of the small scale power spectrum in the forest as a function of position across each spectrum. The Morlet filter is a plane wave, multiplied by a Gaussian and in configuration space may be written as:

x2 Ψ (x)= Kexp(ik x)exp . (4.6) n 0 −2s2  n  The Fourier space counterpart, when the normalization constant K is fixed so the filter has unit power (see Lidz et al. 86) is:

2πs (k k )2s2 Ψ (k)= π−1/4 n exp − 0 n . (4.7) n ∆u − 2 r  

In the above equation, ∆u is the size of each spectral pixel in velocity units and sn is a suitable smoothing scale (also in velocity units) chosen to extract the small scale power, and we set k0sn = 6 (see Lidz et al. 86). Each mock spectrum is convolved with the above filter. We work with the transmission fluctuation field, δ (x) = (F (x) F )/ F where F − h i h i F = e−τ is the transmission and F is the ensemble-averaged mean transmitted flux. h i The transmission fluctuation, convolved with the wavelet filter, is:

a (x)= dx′Ψ (x x′)δ (x′), (4.8) n n − F Z The amplitude of this filtered field, at position “x” is given by

A(x)= a (x) 2, (4.9) | n | and characterizes the amount of small-scale structure in the transmission field. We generally smooth this field with a top-hat of length L, 1 ∞ A (x)= dx′Θ( x x′ ; L/2)A(x′), (4.10) L L | − | Z−∞

171 4.5 Measuring the Temperature of the z 5 IGM ∼

where Θ is a top-hat function. The quantity AL(x) is a measure of the average small scale power across different portions of a quasar spectrum, and should broadly the

temperature of corresponding regions in the IGM, with cold regions giving a larger AL than hot regions.

4.5.1 Hydrodynamic Simulations: Perfect Temperate-Density Relation Models

As a first test, we take high redshift outputs from the hydrodynamic simulation (see 4.2) § and impose temperature-density relations before producing mock quasar spectra. This test ignores the impact of inhomogeneous reionization, but it nonetheless provides some intuition for how well our approach can constrain the IGM temperature. Fig. 4.10 shows an example sightline, 50 Mpc/h in length, extracted from the hydro- dynamic simulation at z = 5 for each of two different temperature-density relation models.

The top panel shows the transmission fluctuation, δF , for models with γ = 1.3 and each of T = 7.5 103 K and T = 2.5 104 K while the bottom panel shows the smoothed 0 × 0 × wavelet amplitudes, AL, in each model. In this case, the smoothing scale sn is set to sn = 51 km/s, the pixel size to ∆u = 3.2 km/s, and L = 1, 000 km/s. In each case the intensity of the ionizing background has been renormalized so that the global mean transmitted flux is F = 0.20. This is the mean transmitted flux implied by extrapolating the recent best-fit h i measurement of Becker et al. (10) to z = 5.1

Although the differences between δF along the two example sightlines are generally small, there are some noticeable differences. In particular, it appears that the “spikes” of transmission in the colder model are more prominent. This is mostly a result of the larger Doppler widths in the hot model. At this redshift, the heights of the transmission

1Specifically, we use these authors’ smooth functional fit to their measured effective optical depth. This is an (approximate) fit to measurements in bins centered on redshifts from z = 2.15 to z = 4.85, and so our extrapolation of this fit out to z = 5 is only very slight.

172 4.5 Measuring the Temperature of the z 5 IGM ∼ spikes are often influenced by nearby gas elements that are centered on saturated or highly absorbed parts of the spectrum; the broad Doppler wings from this gas extend into adjoining unsaturated regions and thereby reduce the height of neighboring transmission spikes. The spikes are less impacted by the narrower Doppler wings in the colder model and remain more prominent. Essentially, the forest has become “inverted” at these redshifts in comparison to at lower redshift. At sufficiently low redshift, the forest is mostly transmitted with some prominent absorption lines interspersed. In the low redshift case most of the information about the IGM temperature comes from narrow absorption lines. At high redshift, the forest is mostly absorbed and most of the information about the IGM temperature is instead in the transmission spikes. In either case, the amount of small scale power in the Ly-α forest is indicative of the temperature of the gas in the IGM. This is illustrated by the bottom panel of Fig. 4.10 for the high redshift case considered here. This panel shows the smoothed wavelet amplitudes along each line of sight. The smoothed wavelet amplitudes are larger in the cold IGM model, with the largest differences occurring near ∆v 4500 km/s, close to several prominent ∼ transmission spikes in the models. One possible complication is that long completely saturated regions will – regardless of temperature – have low wavelet amplitudes, AL, since there is no small scale structure in such regions. These saturated zones may in fact be more prominent in models with low temperature since cold regions recombine more quickly, and hence have larger neutral fractions and suffer more absorption than hot regions.1 Fig. 4.10 suggests, however, that this is not a big effect at z = 5, F = 0.2, although the saturated regions will be more h i prominent at higher redshift (see 4.5.3). We can guard against “contamination” from § saturated regions by masking them before measuring the probability distribution of the wavelet amplitudes, and by varying the smoothing scale L.

1Although the precise impact of the temperature on the wavelet amplitude PDFs shown here is not this transparent since we are comparing models at fixed mean transmitted flux.

173 4.5 Measuring the Temperature of the z 5 IGM ∼

To characterize the variations of the wavelet amplitudes with temperature more quanti- tatively, we calculate the probability distribution function (PDF) of wavelet amplitudes for ensembles of mock spectra generated from different temperature-density relation models. The results of these calculations are shown in Fig. 4.11. The PDF is quite sensitive to the temperature at mean density in these models. For example, the location of the peak in the wavelet amplitude PDF is at an AL that is roughly three times larger in the coldest model shown (with T = 7.5 103 K), compared to the hottest model considered here 0 × (T = 2.5 104 K). For the mean transmitted flux ( F = 0.20) and z = 5, the wavelet 0 × h i PDF for sn = 51 km/s is sensitive mostly to densities near the cosmic mean. As a result,

we find that the wavelet PDFs here depend strongly on T0, but are insensitive to γ, the slope of the temperature-density relation. It is hence important to keep in mind that our approach for measuring the IGM temperature is only sensitive to the temperature of the IGM close to the mean density, and it is therefore not possible to extract the full trend of temperature with density shown in our models (e.g., Fig. 4.5).

Although the PDFs depend sensitively on T0, it is also clear that the wavelet amplitudes are not perfect indicators of the temperature. In the limit that the temperature at mean

density were the only quantity that determined AL, these PDFs should approach delta

functions in AL. That the wavelet PDFs have some breadth is not, however, surprising: the temperature is clearly not the only quantity that determines the small-scale structure in the forest. That said, Fig. 4.11 looks promising and helps to motivate further study.

4.5.2 Degeneracy with the Mean Transmitted Flux

One other potential issue, however, is that the wavelet PDF is also sensitive to the some- what uncertain value of the mean transmitted flux, F . Although the present statistical h i uncertainties on this quantity are σ / F 10% near z 5 (Becker et al. 10), the sys- hF i h i ≤ ∼ tematic uncertainties are significantly larger. In particular, it is difficult to estimate the unabsorbed quasar continuum level, especially at the redshifts of interest for this study,

174 4.5 Measuring the Temperature of the z 5 IGM ∼ where the absorption in the forest is very large (e.g. Faucher-Gigu`ere et al. 54). The mea- surement of the wavelet PDF itself should, however, be fairly robust to uncertainties in the level of the unabsorbed quasar continuum. This is the case because we consider the statistics of the transmission fluctuations, δ = (F F )/ F , for which a (multiplicative) F − h i h i error in the continuum normalization divides out (see Lidz et al. (86) for a discussion and tests with lower redshift data). Nevertheless, we still need to know the mean transmitted flux very accurately: we use this measurement to in turn fix the intensity of the ionizing background at the redshifts of interest, which is itself quite uncertain. The amount of small scale structure in the forest, and hence the model wavelet amplitudes, do depend on the overall mean transmitted flux. As a result, while we should be able to measure the wavelet PDF without knowing the precise continuum normalization, our interpretation of this measurement still requires knowing the mean transmitted flux. To illustrate this, we plot (in the top panel of Fig. 4.12) the z = 5 wavelet amplitude PDF in a model with T = 1.5 104 K and the preferred mean 0 × transmitted flux at this redshift ( F = 0.20, red short-dashed line). As in Fig. 4.11, the h i wavelet amplitudes are smaller in this model than in, for example, the cooler model with T = 7.5 103 K at the same value of the mean transmitted flux (blue long-dashed line in 0 × the top panel of Fig. 4.12). However, if we allow the mean transmitted flux to increase in the colder model, the resulting wavelet PDF becomes similar to that in the hotter model. In particular, the black solid line shows a colder model with the mean transmitted flux increased to F = 0.30; this closely matches the wavelet PDF in the hotter model at the h i smaller mean transmitted flux ( F = 0.20). This particular value of the mean transmitted h i flux, F = 0.30, is well outside the presently allowed range, given the statistical errors h i on current measurements. Nevertheless, the wavelet PDF clearly shows some degeneracy between variations in T and in F . This invites further attention, especially given the 0 h i systematic concerns associated with estimating the unabsorbed continuum level. One way to help break this degeneracy is to combine the measured wavelet PDF with

175 4.5 Measuring the Temperature of the z 5 IGM ∼ a measurement of the flux power spectrum on larger scales. This quantity is especially sensitive to the mean transmitted flux, and the power spectrum of δF has the virtue – like the wavelet PDF – that it is insensitive to the overall normalization of the quasar continuum. On the other hand, on sufficiently large scales, the gentle fluctuations in the underlying quasar continuum still likely contaminate this measurement. However, there should still be a useful range of scales where the structure in the forest dominates over that in the continuum (see e.g. McDonald et al. 103) and it is these scales that we will consider to help break the T F degeneracy. 0 − h i To illustrate how the flux power measurement may help break this degeneracy, we 2 plot the amplitude of transmission fluctuations, ∆F = kPF (k)/π, for a single example wavenumber (k = 0.003 s/km in velocity units, or k = 0.39h Mpc−1 in co-moving units at

z = 5) as a function of mean transmitted flux. The 1-D flux power spectrum (PF (k)) is fairly flat on large scales and so the precise k considered here is not especially important. The bottom panel of Fig. 4.12 shows the flux power spectrum at k = 0.003 s/km as a function of F for the two values of T . In each case, ∆2 is a strong function of F . The h i 0 F h i red triangle and black pentagon show the power spectra in the T = 1.5 104 K and the 0 × T = 7.5 103 K models respectively, for the values of the mean transmitted flux( F = 0.20 0 × h i and F = 0.30) at which their wavelet PDFs are degenerate. The large scale flux power h i spectra in these two models differ by the sizable factor of 1.7. It should be straightforward to measure the flux power spectrum on these scales to this level of accuracy, and so this measurement can help pin down F and break the degeneracy. h i 2 One possible concern with this approach is that the precise relationship between ∆F and F may be somewhat model dependent, and our inability to perfectly model the h i forest – especially at high redshifts, potentially close to the EoR – might lead us to draw spurious conclusions. At present, the only way to guard against this possibility is to test the goodness-of-fit of our models for as wide a range of empirical tests as possible. Ideally, one would compare models with measurements of the flux power across a wide range of scales

176 4.5 Measuring the Temperature of the z 5 IGM ∼

(although significantly larger scales will be subject to contamination from power in the quasar continuum), the wavelet PDF, the mean transmitted flux, and perhaps the statistics of the Ly-β forest as well (e.g. Dijkstra et al. 44, Furlanetto and Oh 58).

4.5.3 Wavelet Amplitude PDFs in Inhomogeneous Reionization Models

With the results of the previous section as a guide, we now turn to consider the wavelet amplitude PDFs in the more realistic inhomogeneous temperature models developed in 4.4. § In this case, we are using the dark matter simulations of McQuinn et al. (108) along with our model temperature distributions. Although the large volume of these simulations allows us to capture the reionization-induced inhomogeneities, they are not – taken as is – adequate for capturing the small-scale structure in the Ly-α forest, which is the basic observable we aim to explore here. In order to make headway, we add small-scale structure to sightlines extracted from the simulation cube using the log-normal model, as in Kohler et al. (78). Briefly, we gen- erate one-dimensional Gaussian random fields δG using the one-dimensional linear density power spectrum (scaled to the redshift of interest, and calculated after smoothing the three- −2k2/k2 −1 dimensional linear power spectrum with a filter of the form e f and kf = 30h Mpc to loosely mimic the effect of Jeans smoothing, Gnedin and Hui 64). From the Gaussian random realizations, we produce lognormal fields at high resolution using the transforma- 2 δG−σ /2 2 tion 1 + δLN = e G , where σG is the variance of the Gaussian random field. As in Kohler et al. (78) the lognormal field is modulated by the larger scale modes captured in the simulation (δsim) according to 1 + δ = (1+ δsim)(1 + δLN) with the (subscript-free) symbol δ denoting the density contrast with added small-scale structure. Similarly, using the simulated temperature in a coarse pixel (described by T0 and γ), the temperature in a γ−1 fine pixel becomes T = T0(1 + δ) . The main disadvantage here is that the resulting sightlines have too much large scale structure: the lognormal field adds both large and small scale modes to the simulation,

177 4.5 Measuring the Temperature of the z 5 IGM ∼ and the simulation was not deficient in large scale power to begin with. This is partly mitigated by our using a slightly higher redshift simulation output (z = 6.9) than the redshift of interest. This simple approach is hence imperfect, but the added small scale structure does nevertheless allow us to reliably model the impact of thermal broadening on the resulting mock Ly-α forest spectra. As a test, we measure the flux power spectrum from the mock “lognormal-enhanced” sightlines and compare them with the flux power spectrum from mock spectra generated from the hydrodynamic simulation. The flux power spectrum from the lognormal spectra is roughly 50% larger than from the hydrodyamic simulations. However, the overall shape of the flux power is fairly well captured in the lognormal case, and importantly, the shape of the flux power spectrum varies in a similar way in both calculations as the temperature and thermal broadening are varied. Hence we believe that this approach suffices to capture the main impact of patchy reionization on the small scale structure in the Ly-α forest. We caution, however, that a detailed comparison with upcoming measurements will certainly require improvements here. With this cautionary remark, we turn to consider the properties of mock spectra drawn from our inhomogeneous temperature models. Fig. 4.13 shows typical example sightlines from the High-z and Low-z reionization models at z = 5.5 and F = 0.1. The trends are h i broadly similar to those in Fig. 4.10: the colder models have more small scale structure than the hotter models. As a result, the transmission field has more prominent spikes in the colder High-z model, and the smoothed wavelet amplitudes in this model (bottom panel) are larger than in the Low-z model. The inhomogeneous models incorporate, however, the reionization-induced temperature variations that are not included in the previous model, although the impact of these variations are generally hard to discern by eye. One can however identify that the prominent cold region in the middle of this sightline, for example, corresponds to a pronounced peak in the smoothed wavelet amplitude field in each model.

Note that the wavelet amplitudes and δF fluctuations are larger here than in Fig. 4.10 because here we consider z = 5.5 and F = 0.1, while in the previous figure we considered h i

178 4.5 Measuring the Temperature of the z 5 IGM ∼ z = 5.0 and F = 0.2. In addition, the lower transmitted flux considered here leads to more h i completely absorbed regions in the mock Ly-α forest, and these regions have correspondingly low wavelet amplitudes. As discussed earlier, these regions do not contain information about the IGM temperature. More quantitatively, the resulting wavelet amplitude PDFs for some example inhomo- geneous temperature models are shown in Fig. 4.14. Since the mean transmitted flux at this redshift (z = 5.5) is somewhat uncertain, we compare models normalized to each of F = 0.2 (top panel) and F = 0.1 (bottom panel) in order to illustrate the impact h i h i of varying F . Extrapolating the best fit measurement from (Becker et al. 10), we find h i F = 0.13 at z = 5.5 so this range should approximately bracket the expected value, al- h i though the lower end of this range is preferred. Note that there is some evidence that the mean transmitted flux decreases more rapidly above z 5.5 or so compared to the evolution ∼ expected from lower redshifts (e.g. Fan et al. 52), and so this would further favor the lower value, and perhaps even slightly smaller numbers than considered here. Nevertheless, it is worth considering the mean transmitted flux dependence explicitly: even if F = 0.2 is h i reached at z = 5 rather than z = 5.5, the temperature fluctuations could be as large at z = 5 as in our z = 5.5 models if reionization is more extended than considered here.

In each case, the wavelet filter’s smoothing scales are set to sn = 34 km/s and L = 1, 000 km/s, while the pixel size is ∆u = 2.1 km/s.1 In each panel, the PDF in the High-z model is compared to Low-z models for two different reionization temperatures, T = 2 104 K r × and T = 3 104 K. In addition, we plot the wavelet amplitude PDF for a model with a r × completely homogeneous temperature field. In this isothermal case the temperature is set to T = 1.44 104 K, matching the median temperature near the cosmic mean density in 0 × the Low-z, T = 3 104 K model. Homogeneous reionization models with the same T r × 0 but differing γ would give very similar results, since the wavelet amplitudes are sensitive to

1The pixel size here matches that of typical Keck HIRES spectral pixels. Note that the smoothing scale, sn, and pixel size, ∆u, here are similar, but slightly different than used in the previous section. The large scale smoothing, L, is identical.

179 4.5 Measuring the Temperature of the z 5 IGM ∼ absorbing gas near the cosmic mean density for these values of the mean transmitted flux. Comparing the T = 1.44 104 K isothermal model with the Low-z, T = 3 104 K case 0 × r × helps to illustrate the impact of temperature inhomogeneities. First, we consider how the location of the peak in the PDF varies with the reionization model. The qualitative behavior is as expected: the High-z model peaks at the largest wavelet amplitude, while the Low-z models peak at smaller wavelet amplitude. This results because the High-z model is colder than the Low-z models, and so it has more small- scale structure and hence higher wavelet amplitudes. The Low-z model with the higher reionization temperature, T = 3 104 K, has hotter gas at the redshift of interest (z = 5.5) r × than in the Low-z scenario with T = 2 104 K and so the PDF in the former model is r × peaked at smaller wavelet amplitudes. These trends occur in both the F = 0.1 and F = 0.2 models. In the smaller F h i h i h i model the PDFs peak at larger A , reflecting the enhanced power in the δ = (F F )/ F L F −h i h i field for decreasing mean flux. To provide a quantitative comparison, the High-z model has an average wavelet amplitude that is 73% larger than in the Low-z, T = 2 104 K model, r × while the average amplitude in the T = 3 104 K, Low-z model is 27% smaller than the r × T = 2 104 K case. These numbers are for F = 0.2, but the fractional differences are r × h i similar for F = 0.1. h i Next we consider the width of the wavelet amplitude PDFs. The width arises in part

because AL is not a perfect tracer of temperature, and also because the temperature field is inhomogeneous. The latter contribution to the width can in principle be used to constrain the spread in the timing of reionization, and so this quantity is highly interesting. Compar- ing first the inhomogeneous models in Fig. 4.14, it is clear that the Low-z, T = 3 104 K r × model has the widest distribution of wavelet amplitudes, followed by the Low-z, T = 2 104 r × K model, while the High-z model has the narrowest AL distribution of these models. This is expected, since the High-z model has the smallest temperature fluctuations, while the Low-z, T = 3 104 K model has the largest temperature fluctuations. It is also instruc- r ×

180 4.5 Measuring the Temperature of the z 5 IGM ∼ tive to compare the isothermal models (magenta dot-dashed lines in each panel) with the Low-z, T = 3 104 K models. The isothermal model has the same median temperature r × near the cosmic mean density as the Low-z, T = 3 104 K model. This PDF is similar, r × but narrower, than in the inhomogeneous temperature case. Quantitatively, the fractional width of the distribution, σ / A , is 18% larger in the Low-z, T = 3 104 K model A,L h Li r × than in the homogeneous case at F = 0.2 and 11% larger at F = 0.1. These relatively h i h i small differences seem challenging to extract, but are sufficiently interesting to merit further investigation.

4.5.4 Forecasts

Finally, we briefly forecast the significance at which various models may be distinguished using existing data samples. Here we will be content with rough estimates. For simplicity, we predict the expected error bar on only the first two moments of the wavelet amplitude PDF, and compare this to the difference between some of our models. We consider a sample of Nlos independent spectra, and assume that each spectrum has sufficient S/N so that we can estimate error bars in the sample variance limit – i.e., we work in the limit that photon noise from the night sky and the quasar itself, as well as instrumental noise, are negligible compared to sample variance (also termed “”). Our error budget hence reflects the scatter expected – given the large scale structure of the universe and the limited volume probed by our hypothetical survey – around the true value that would be obtained if we could average over an infinite volume. It is instructive to first consider the type of quasar spectra that are required to measure the wavelet amplitudes in the sample variance limit. The first obvious requirement is that the spectral resolution needs to be high enough to resolve the thermal broadening scale, which is on the order of 10 km/s. This can be achieved with, for example, Keck HIRES ∼ spectra which have a spectral resolution of FWHM = 6.7 km/s. Spectra from the MIKE spectrograph on would partly resolve the thermal broadening scale: the resolution

181 4.5 Measuring the Temperature of the z 5 IGM ∼ of these spectra is a factor of 2 worse than HIRES (e.g. Becker et al. 8). Next, we consider ∼ the impact of photon and instrumental noise. In particular, we estimate the S/N (at the continuum) per HIRES pixel at which the expected shot-noise is a small fraction of the average wavelet amplitude in plausible models. The mean wavelet amplitude from the noise should be roughly A (N/S)2/ F (Hui et al. 71, Lidz et al. 86). In order for the noise h noisei≈ h i to be sub-dominant, we impose that the noise should be less than 10% of the mean wavelet amplitude in our Low-z, T = 2 104 K model at F = 0.2 (which has A = 0.35). This r × h i h i requires a S/N & 12 at the continuum, per 2.1 km/s HIRES pixel. This is a fairly stringent requirement for quasars at the high redshifts of interest for our proposed measurements, but this sensitivity has been reached already in previous work. We could likely make a less stringent requirement on the S/N of the data sample: this would just necessitate careful shot-noise subtraction, and boost our error budget somewhat. Currently, we are aware of roughly 10 HIRES spectra in the published literature at z 5 5.7 that meet our S/N ∼ ∼ − and resolution criteria (e.g. Becker et al. 12). There are substantially larger numbers of lower resolution and S/N spectra from the SDSS that could potentially be followed-up at higher resolution and sensitivity to improve the statistics here. For example, there are 36 SDSS-DR7 quasars in the z = [5.0, 5.2] redshift bin of Becker et al. (10). We now proceed to estimate the sample variance errors. The first quantity of interest is the sightline-to-sightline scatter in the wavelet amplitudes, averaged over the entire Ly-α forest region of each quasar. We define PA,L(k) to be the power spectrum of the fluctuations in the wavelet amplitude after smoothing the amplitudes on scale L, i.e., the power spectrum of δ (x) = (A (x) A )/ A . We relate the expected error bars to this power spectrum, A,L L −h Li h Li and estimate them by measuring PA,L(k) from our simulated models. This approach has the advantage that we can approximately extrapolate PA,L(k) to scales beyond that of our simulation box and roughly account for missing large scale Fourier modes. This power spectrum of δA,L(x) is a four point function of the flux and is related to the power spectrum

182 4.5 Measuring the Temperature of the z 5 IGM ∼ of A(x) (the unsmoothed wavelet amplitude power spectrum) from Eqs. 4.9 and 4.10 by

sin(kL/2) 2 P (k)= P (k). (4.11) A,L kL/2 A   This is just a filtered version of the wavelet amplitude power spectrum. From each indepen- dent sightline, we estimate the moments of the wavelet amplitude PDF by further averaging

over a length scale Lspec, comparable to the separation (in velocity units) between the Ly-α and Ly-β emission lines from the quasar. Note the distinction between the two smooth- ing scales here: L is the smoothing scale over which we are studying the wavelet amplitude

variations, while Lspec is the scale over which we estimate the moments from each spectrum. The formula for the sample variance for a single sightline is then (Lidz et al. 86):

2 ∞ ′ 2 σ (Lspec) dk′ sin(k L /2) A,L = spec P (k′). A 2 2π k′L /2 A,L h i Z−∞  spec  (4.12)

In a sample of Nlos independent Ly-α forest sightlines, the expected (1-σ) fractional error on A (in the sample variance limit) is: h Li δ A 1 σ (L ) h Li = A,L spec . (4.13) A √N A h Li los h Li We can compare this estimate of the fractional error in the average wavelet amplitude with the difference between the average amplitudes in some of the models of the previous section. Next, we want to consider the expected error bar on the second moment of the wavelet PDF. This second moment provides one diagnostic for the impact of temperature inho- mogeneities from patchy reionization. We would like to check whether the variance of the

AL distribution is broad enough to imply patchy reionization. This requires computing 2 the “variance of the variance”; in particular, we want the variance of an estimate of σA,L

when this quantity is estimated from a sightline of length Lspec. We calculate this quantity

183 4.5 Measuring the Temperature of the z 5 IGM ∼

assuming that AL approximately obeys Gaussian statistics. In this case, one can show that the desired variance is: ∞ dk′ sin(k′L /2) 2 Var[σ2 (L )] =2 A 4 spec A,L spec h Li 2π k′L /2 Z−∞  spec  ∞ dk′′ P (k′ k′′)P (k′′) × 2π A,L − A,L Z−∞ + 4 A 4σ2 (L ). (4.14) h Li A,L spec

2 The expected error on an estimate of σA,L from a sample of Nlos independent sightlines is then:

2 2 δσ 1 Var[σA,L(Lspec)] A,L = . (4.15) 2 q 2 σA,L √Nlos σA,L We can now plug numbers into Eqs. 4.13 and 4.15 to estimate the ability of current samples to constrain some of our models. We assume that Nlos = 10 sightlines are available for our study, and take z = 5.5, F = 0.2 here. We assume the true model is the T = 3 104 h i r × K, Low-z case and examine at what significance other models may be distinguished from this case. Conservatively, we assume that L = 2.5 104 km/s; the velocity separation spec × between the Ly-α and Ly-β emission lines is 5.1 104 km/s, and so our assumed value × effectively masks-out half of the forest. While one will want to mask-out proximity regions, damped Ly-α systems, prominent metal lines, etc., our choice is certainly conservative. In this case, evaluating Eq. 4.13 in our assumed true model, we find that a measurement of A from N = 10 sightlines should rule out the High-z (T = 2 104 K) model at 32 σ, h Li los r × − and a Low-z model with the lower reionization temperature (T = 2 104 K) at 9 σ! These r × − forecasts are optimistic, since we have assumed – for example – perfect knowledge of the mean transmitted flux; in practice, the mean transmitted flux may have to be constrained separately from the large-scale flux power spectrum ( 4.5.2). Nonetheless, we believe the § overall point is robust: existing samples should provide interesting constraints on A . h Li

184 4.6 Conclusions

Significantly more challenging is to measure the variance of the AL distribution well enough to identify signatures of patchy reionization. With the optimistic “true” model considered here (the Low-z, T = 3 104 K case), however, we forecast that N = 10 r × los spectra are sufficient to rule out the homogeneous T = 1.44 104 K model1 at 2.4 σ, × − based on the variance of the AL distribution alone. Note that both of these estimates assumed F = 0.2 and our z = 5.5 temperature models but the expected constraints are h i similar for F = 0.1. h i

4.6 Conclusions

In this work, we modeled the temperature of the IGM at z & 5, incorporating the impact of spatial variations in the timing of reionization across the universe. We contrasted the z 5 temperature in models where reionization completes at high redshift – near z = ∼ 10 – with scenarios where reionization completes later, near z = 6. In agreement with previous work (Furlanetto and Oh 58, Trac et al. 177)2, we found that the properties of the z = 5 temperature differ markedly between these two models. The IGM is cooler in the early reionization model, and the usual temperature-density relation is a good description of the temperature state in this case, while the temperature state is more complex and inhomogeneous in the late reionization scenario. We then produced mock z & 5 Ly-α forest spectra from our numerical models, in effort to explore the observable implications of the IGM temperature as close as possible to hydrogen reionization. In particular, we used the Morlet wavelet filter approach of Lidz et al. (86) to extract the small-scale structure across each Ly-α forest spectrum. The small- scale structure in the forest is sensitive to the temperature of the IGM, and the filter we

1Recall that the temperature in this model matches the median temperature for gas near the cosmic 4 mean density in the Low-z, Tr = 3 × 10 K model. 2This is also in general agreement with still earlier work by Theuns et al. 171 and Hui and Haiman 74, although these two studies did not incorporate inhomogeneities in the timing of reionization.

185 4.6 Conclusions

use is localized in configuration space, which makes it well-suited for application in cases where the temperature field is inhomogeneous. Interestingly, we found that the small-scale structure in the forest is sensitive to the IGM temperature even when the forest is highly absorbed. In particular, the transmission field in between absorbed regions is more spiky if the IGM is cold, compared to hotter models. Using existing high resolution Ly-α forest samples, one should be able to use this difference to distinguish between high redshift and lower redshift reionization models at high significance. It may, however, be necessary to combine measurements of the small- scale structure in the forest with measurements of the larger scale flux power spectrum to help break degeneracies with the mean transmitted flux, which is hard to estimate directly at the high redshifts of interest for these studies. In addition, we considered the impact of spatial variations in the timing of reionization on the width of the wavelet amplitude distribution. We found that these variations broaden the width of this distribution, but that the broadening is fairly subtle. This likely results in part because the temperature variations we are interested in are coherent on rather large scales, and aliasing – from fluctuations in the transmission field transverse to the line of sight – obscures our ability to measure large scale fluctuations along the line of sight (e.g. Lai et al. 82, McQuinn et al. 107). Nonetheless, we forecast that our Low-z, T = 3 104 K r × model can be distinguished from a homogeneous temperature model at 2 3σ with existing − samples of ten high resolution sightlines. Larger samples could improve on this, and an analysis of the small-scale structure in the Ly-β forest might help as well. In this chapter, we focused on the small-scale structure since it is a direct indicator of the temperature, but another approach would be to consider instead transmission fluctuations on large scales, especially as probed in “3D” measurements of the Ly-α forest (e.g. McQuinn et al. 107). This may be possible at z & 4 with DESI (Levi et al. 84, McQuinn et al. 107). To robustly interpret future measurements, our modeling should be improved in var- ious ways. In particular, we should incorporate inhomogeneous Jeans smoothing effects

186 4.6 Conclusions

into our modeling. This might be accomplished by, for example, incorporating our semi- numeric modeling on top of a large dynamic range HPM (Gnedin and Hui 64) simulation. These calculations will need to face the competing requirements of capturing the large-scale variations in the timing of reionization, while simultaneously resolving the filtering scale. Nevertheless, we believe that measurements of the z & 5 IGM temperature should provide a valuable handle on the reionization history of the universe.

Appendix: Approximate Thermal History Calculations

Here we derive an approximate analytic formula for the thermal history of an IGM gas element using linear perturbation theory. Here our derivation is quite similar to the analytic calculation in Hui and Gnedin (73) (their 3.1), except here we include Compton cooling § off of the CMB, which is important for our application in which we consider high redshift reionization and the temperature at redshifts close to reionization. It is instructive to first briefly examine which heating/cooling processes are important, in addition to the usual adiabatic heating/cooling from the contraction/expansion of gas parcels. In particular, Fig. 4.15 compares the relative importance of HI photoheating, HeI photoheating, Compton cooling, HII recombination cooling, HeII recombination cooling, and free-free emission cooling for intergalactic gas at z 7. The figure assumes that the ∼ gas is in ionization equilibrium and that hydrogen is highly ionized and helium mostly singly ionized. As in the body of the text, we are assuming that HeII reionization has not yet commenced at the redshifts of interest. For the highly-ionized and low density intergalactic gas considered here, line excitation cooling and collisional ionizations should be unimportant. The left hand panel considers the various heating and cooling processes for gas of fixed temperature, T = 104 K, as a function of density while the right hand panel shows the same for gas at the comic mean density as a function of temperature. In each panel, the curves in the figure indicate the absolute values of the various rates, so that cooling processes are shown as positive numbers on the plot. The photoheating curves

187 4.6 Conclusions

assume that the hardened ionizing spectrum follows a J(ν) ν−1.5 power-law near the ∝ photoionization edges. The dominant processes are clearly HI photoheating and Compton cooling. After these processes in importance are HeI photoheating and HII recombination cooling: these have rates that are roughly 20% smaller than HI photoheating near the cosmic mean density and T 104 K at z 7. It is also interesting to note that at the ∼ ∼ densities considered here and for the adopted ionizing spectrum, HeI photoheating and HII recombination cooling have nearly equal magnitudes near T 104 K; since these two ∼ processes enter the thermal evolution equation with opposite signs, this leads to a partial cancellation. As a result, a good approximation to the IGM thermal evolution (at high redshifts before HeII reionization) results from including only adiabatic heating/cooling, Compton cooling, and HI photoheating. Note, however, that in the body of the work we include all of the additional processes considered in Fig. 4.15. The approximate results here are nonetheless useful and fairly accurate, and can in turn help to build intuition. The approximate equation for the thermal evolution is then: −0.7 4 dT 2T dδ α n¯ E T 4 σT aradT = 2HT + + 0 e J (1 + δ)+ γ (T T ) . dt − 3(1 + δ) dt 3(1 + χ )k 104K 3 m c γ − He B   e (4.16)

4 1 Here α0 is the (case-A) recombination coefficient for hydrogen at T = 10 K , EJ is the 2 average energy injected into the gas per photoionization , Tγ is the CMB temperature (at 4 the scale factor of interest), σT is the Thomson scattering cross section, and aradTγ is the energy density in the CMB. The first two terms describe adiabatic cooling/heating, the third term is from photoionization heating, and the last term accounts for Compton cooling. This equation assumes that the gas is in photoionization equilibrium, adopts an electron number

1 We assume the case-A recombination coefficient and use the approximation αA = 4.2 × 10−13(T/104K)−0.7 cm3s−1 (Hui and Gnedin 73) in this Appendix. 2 −α Assuming a power-law spectral index, Jν ∝ ν (with the power law accounting for hardening from absorption), EJ = hνHI /(α + 2).

188 4.6 Conclusions

density of ne = nH + nHe, and assumes the number density of free particles in the gas is

ntot = ne + nH + nHe = 2(nH + nHe). γ−1 As in Hui and Gnedin (73), we assume a solution of the form T = T0(1 + δ) and linearize (T T [1 + (γ 1)δ]) to find equations for T and γ 1, as functions of scale ≈ 0 − 0 − factor. Let us first introduce two constants to make the notation more compact: α n¯ (0)E A = (104K)0.7 0 e J . (4.17) 3(1 + χHe)kBH0√Ωm

Heren ¯e(0) denotes the present day (z = 0), spatially averaged, electron number density. Note that the constant A has dimensions of [A] = [K]1.7. In our fiducial model with α = 1.5, the numerical value of A is A = 5.77 105K1.7. Second, we introduce × B 1 3mec = ; tComp = 4 . (4.18) H0√ΩmtComp(0) 4σT aradTγ

Here tcomp(0) a characteristic timescale for Compton cooling today (z = 0); this timescale falls off towards high redshift as t a4. In our assumed cosmology, the numerical value comp ∝ of this constant is B = 1.15 10−2. × Using the high redshift approximation for the Hubble parameter, H H √Ω a−3/2, ≈ 0 m and setting δ = 0 to find an equation for T0(a) valid in linear theory, Eq. 4.16 gives the following equation for T0:

d(a2T ) 0 = Aa0.9(a2T )−0.7 Ba−7/2(a2T )+ BT (0)a−5/2. (4.19) da 0 − 0 γ A similar equation follows for γ 1 (again valid to linear order in δ, and with the approxi- − mations to the thermal evolution equation in Eq. 4.16):

d(γ 1) 2 1 BT (0) − = (γ 1) + Aa0.9(a2T )−1.7 [1 1.7(γ 1)] γ a−5/2(γ 1)(4.20). da 3 − − a 0 − − − a2T −   0

We can safely neglect the third term in Eq. 4.19, and we find a solution for T0(a) of the form

(with the initial condition that the gas element is ionized at scale factor ar to a temperature

189 4.6 Conclusions

Tr):

tr B −5/2 ′ u = u exp 0.68Ba−5/2 0.68Ba−5/2 + 0.68A(0.68B)0.76e0.68 a dt′t′−1.76e−t . r − r Zt h 2 1.7 i 2 1.7 B −5/2 B −5/2 u = (a T0) ; ur = (arTr) ; t = 0.68 a ; tr = 0.68 ar (4.21).

The corresponding solution to Eq. 4.20 for the evolution of γ does not have a simple closed analytic form, but the equations can be solved numerically. Comparing the solutions for T0(z) from Eq. 4.21 and γ(z) from Eq. 4.20 with results given in the body of the text, we find that the approximate solutions are good to better than 10% accuracy. This accuracy is, in fact, somewhat better than might be expected given the approximations made, and may reflect cancelations between some of the neglected terms (such as the compensating omissions of HeI photoheating and HII recombination cooling, as highlighted in Fig. 4.15). Nevertheless, the approximate solutions seem quite useful and so we include them here.

190 4.6 Conclusions

Figure 4.8: Thermal state at z =5.5 for various reionization temperature and spectral shape models. This is similar to the z = 5.5 curves in Fig. 4.2, except here we vary the reionization temperature, Tr, and the spectral shape, α. Increasing Tr leads to a higher T0 and a flatter γ for recently reionized gas parcels, while parcels that reionize at sufficiently high redshifts are insensitive to Tr. A harder ionizing spectrum after reionization (smaller α) leads mostly to a slightly larger value of the asymptotic temperature achieved at high zr. The harder spectrum also slightly hastens the transition of γ to its asymptotic value.

191 4.6 Conclusions

Figure 4.9: Temperature density relation at z = 5.5 for various reionization temperatures in the High-z and Low-z models. The “X”s in the legend indicate the color of the points in the corresponding models, while the dashed lines in the same models have different colors to promote visibility. The models in the legend are listed from top to bottom: the highest points and line (indicating the median temperature at various densities) show the T = 2 104 K, r × Low-z model; next is the T =1 104 K, Low-z model; then the T =3 104 K, High-z model; r × r × and finally the T =3 104 K, High-z model. r ×

192 4.6 Conclusions

Figure 4.10: Example sightlines and wavelet amplitudes for two different models of the IGM 4 temperature at z 5. The top panel shows δF (x) for an example sightlines with T0 =2.5 10 ∼ 3 × K, γ = 1.3 (red dashed) and the same sightline except with T0 = 7.5 10 K, γ = 1.3 (black × solid). The bottom panel shows the smoothed wavelet amplitudes, AL, along each spectrum. The lower temperature model has more small scale structure and larger wavelet amplitudes.

The smoothing scale sn = 51 km/s here, while ∆u =3.2 km/s and L =1, 000 km/s.

193 4.6 Conclusions

Figure 4.11: Probability distribution of A for various T0 models at z 5. Each model L ∼ here assumes a perfect temperature density relation with γ = 1.3, and in each case the mean transmitted flux has been fixed – by adjusting the intensity of the ionizing background – to F =0.20. As in Fig. 4.10, the smoothing scale has been set to s = 51 km/s, while ∆u =3.2 h αi n km/s and L =1, 000 km/s.

194 4.6 Conclusions

Figure 4.12: Degeneracy with F . Left panel: Although the PDF of AL is sensitive to T0, this h i 4 effect is degenerate with the impact of varying F . For instance, the model with T0 =1.5 10 h i 3 × K and F = 0.20 is closely mimicked by a colder model with T0 = 7.5 10 K, yet a larger h i × mean transmission of F = 0.30. Right panel: This illustrates that the degeneracy can be h i broken by measuring the (relatively) large scale flux power spectrum. The curves here show the flux power spectrum, evaluated at a single convenient (larger-scale) wavenumber of k = 0.003 s/km, in each T0 model as a function of F . The triangle and pentagon show the flux power for h i each model at the F for which the wavelet amplitude PDFs are degenerate in the two models. h i The large scale flux power in these two models differs appreciably and can be used to break the degeneracy. The red dotted and black dotted horizontal lines are intended only to guide the eye.

195 4.6 Conclusions

Figure 4.13: Example sightlines and wavelet amplitudes from the Low-z and High-z reioniza- tion models. In the models here, the global mean flux is F =0.1 and z =5.5. In each panel h i the red dotted line shows a sightline through the T =3 104 K, Low-z reionization model while r × the black solid line is the same sightline, except in this case the temperature field is drawn from the High-z reionization model (with T = 2 104 K). The simulated density and temperature r × fields have small scale structure added according to the lognormal model, as described in the text. Top panel: The simulated temperature field. Middle panel: The transmission field, δF .

Bottom panel: The smoothed wavelet amplitude with L = 1, 000 km/s, sn = 34 km/s, and ∆u = 2.1 km/s. The transmission fluctuations and wavelet amplitudes are larger than in Fig. 4.10, mostly because of the lower mean transmitted flux adopted here.

196 4.6 Conclusions

Figure 4.14: Probability distribution of AL for various reionization and temperature models at z = 5.5. Left panel: In this panel all models are normalized to F = 0.2. The solid black h i curve shows the wavelet amplitudes for the High-z reionization model (with T = 2 104 K), r × while the red dotted and blue dashed curves show Low-z reionization models with reionization temperatures of T =2 104 K and T =3 104 K respectively. The magenta dot-dashed line r × r × shows a homogeneous temperature model for comparison. In this case, the temperature was set to match the median temperature in the Low-z, T = 3 104 K model for gas at the cosmic r × mean density; the broader distribution in the Low-z model reflects the impact of inhomogeneous reionization. Right panel: Identical to the top panel, but here the models fix F =0.1. In each h i case, the filter scale and pixel size are set to sn = 34 km/s and ∆u = 2.1 km/s respectively, while L =1, 000 km/s.

197 4.6 Conclusions

Figure 4.15: Heating/cooling rates at z 7. Left panel: The (absolute value of) the rates for ∼ relevant processes in the IGM at T = 104 K as a function of density, assuming that hydrogen is highly ionized and that helium is mostly singly-ionized. Right panel: Similar to the left panel except the rates are shown as a function of temperature for gas at the comic mean density.

198 Chapter 5

Identifying Ionized Regions in Noisy Redshifted 21-cm Observations

5.1 Introduction

The Epoch of Reionization (EoR) is the time period when early generations of galaxies first turn on and gradually photoionize neutral hydrogen gas in the surrounding intergalactic medium (IGM). The IGM is expected to resemble a two phase medium during reionization. One phase consists of highly ionized regions, termed ‘ionized bubbles’, that form around clustered groups of ionizing sources, while the other phase is made up of intervening mostly neutral regions that shrink and eventually vanish as reionization progresses. A primary goal of reionization studies is to determine the size distribution and volume-filling factor of the ionized bubbles. This should, in turn, significantly improve our understanding of high redshift galaxy and structure formation. A wide variety of current observations have started to provide tantalizing hints regarding the timing and nature of the EoR (e.g., Fan et al. 52, Totani et al. 176, Dunkley et al. 48, Ouchi et al. 132, Bouwens et al. 23, Mortlock et al.

199 5.1 Introduction

130, Zahn et al. 192, Schenker et al. 162), but we still await a more detailed understanding. A highly anticipated way of improving our knowledge of the EoR is to directly detect intergalactic neutral hydrogen from the EoR using the redshifted 21 cm transition (e.g., Madau et al. 96, Zaldarriaga et al. 194, Furlanetto et al. 60). Indeed, several radio tele- scopes have been constructed, or are presently under construction, in effort to detect this signal, including the Giant Metrewave Radio Telescope (GMRT) (Paciga et al. 135), the Low Frequency Array (LOFAR) (Harker et al. 68), the Murchison Widefield Array (MWA) (Lonsdale et al. 94), and the Precision Array for Probing the Epoch of Reionization (PA- PER) (Parsons et al. 136). This method provides the most direct, and potentially most powerful, way of studying reionization, but several challenges need first to be overcome. In particular, upcoming surveys will need to extract the faint cosmological signal in the pres- ence of strong foreground emission from our own galaxy and extragalactic point sources, and to control systematic effects from the instrumental response, polarization leakage, cal- ibration errors, and other sources of contamination (e.g., Liu et al. 93, Datta et al. 40, Harker et al. 68, Petrovic and Oh 143, Morales et al. 126, Parsons et al. 138). In addition, thermal noise will prevent early generations of 21 cm experiments from making detailed maps of the reionization process. Instead, detections will mostly be of a statistical nature (McQuinn et al. 114). For example, a primary goal of these experiments is to measure the power spectrum of 21 cm brightness temperature fluctuations by binning together many individually noisy Fourier modes (Zaldarriaga et al. 194, Morales and Hewitt 127, Bowman et al. 24, McQuinn et al. 114). It is unclear, however, how best to analyze the upcoming redshifted 21 cm data. Most previous work has focused only on the power spectrum of 21 cm brightness temperature fluctuations (e.g., Furlanetto et al. 61, Lidz et al. 91, Mesinger et al. 119). This statistic does not provide a complete description of the 21 cm signal from the EoR, which will be highly non-Gaussian, with large ionized regions of essentially zero signal intermixed with surrounding neutral regions. The power spectrum, and especially its redshift evolution, do

200 5.1 Introduction

encode interesting information about the volume-averaged ionized fraction and the bubble size distribution (e.g., Lidz et al. 91). However, these inferences are somewhat indirect and likely model dependent, and so it is natural to ask if there are more direct ways of determining the properties of the ionized regions. The approach we explore here is to check whether it may be possible to directly identify ionized regions in noisy redshifted 21 cm observations by applying suitable filters to the noisy data. Our aim here is to blindly identify ionized bubbles across an entire survey volume, rather than to consider targeted searches around special regions, such as those containing known quasars (e.g., Wyithe and Loeb 185, Friedrich et al. 57). Since the 21 cm signal from reionization is expected to have structure on rather large scales – & 30 h−1Mpc co-moving (Furlanetto et al. 62, Iliev et al. 76, Zahn et al. 189, McQuinn et al. 110) – it may be possible to make crude images of the large scale features even in the regime where the signal to noise per resolution element is much less than unity. Even if it is only possible to identify a few unusually large ionized regions in upcoming data sets, this would still be quite valuable. Any such detection would be straightforward to interpret, and would open- up several interesting possibilities for follow-up investigations. Towards this end, we extend previous work by Datta et al. (41) and Datta et al. (42), who considered the prospects for detecting ionized regions using an optimal matched filter. A matched filter is constructed by correlating a known ‘template’ signal with a noisy data set in order to determine whether the template signal is present in the noisy data. Matched filters are used widely in astrophysics: to name just a few examples, matched filters are used to detect clusters in cosmic microwave background (CMB) data (Haehnelt and Tegmark 66), to identify galaxy clusters from weak lensing shear fields (e.g., Hennawi and Spergel 70, Marian et al. 101), and are central to data analysis efforts aimed at detecting gravitational waves (e.g., Owen and Sathyaprakash 133). The outline of this chapter is as follows. In 5.2 we describe the mock 21 cm data sets § used in our investigations. We use the mock data to first consider the ability of future

201 5.2 Method

surveys to make maps of the redshifted 21 cm signal ( 5.3). In 5.4, we then quantify § § the prospects for identifying individual ionized regions using a matched filter technique. In 5.5 and 5.6 we consider variations around our fiducial choice of reionization history § § and redshifted 21 cm survey parameters. We compare with previous related work in 5.7, § and conclude in 5.8. Throughout we consider a ΛCDM cosmology parametrized by n = § s 1,σ8 = 0.8, Ωm = 0.27, ΩΛ = 0.73, Ωb = 0.046, and h = 0.7, (all symbols have their usual meanings), consistent with the latest WMAP constraints from Komatsu et al. (79).

5.2 Method

Briefly, our approach is to construct mock redshifted 21 cm data sets and check whether we can successfully identify known ‘input’ ionized regions in the presence of realistic levels of instrumental noise and the degrading impact of foreground cleaning. Here we describe the ingredients of our mock data sets: our simulations of reionization and the 21 cm signal, our model for thermal noise, and our approach for incorporating the impact of foreground cleaning.

5.2.1 The 21 cm Signal

First, let us describe the underlying 21 cm signal and our reionization simulations. The 21 cm signal will be measured through its contrast with the cosmic microwave background (CMB). The brightness temperature contrast between the CMB and the 21 cm line from

a neutral hydrogen cloud with neutral fraction xHI and fractional baryon overdensity δρ is (Zaldarriaga et al. 194): T T 1+ z 1/2 δT = 28x (1 + δ ) S − γ mK. (5.1) b HI ρ T 10  S   Here Tγ denotes the CMB temperature and TS is the spin temperature of the 21 cm line. Here and throughout we neglect effects from peculiar velocities, which should be a good ap- proximation at the redshifts and neutral fractions of interest (e.g., Mesinger and Furlanetto

202 5.2 Method

118, Mao et al. 99). Furthermore, throughout we assume that the spin temperature is glob- ally much larger than the CMB temperature, i.e., we assume that T T . In this case the S ≫ γ 21 cm signal appears in emission and the brightness temperature contrast is independent of

TS. This is expected to be a good approximation for the volume-averaged ionized fractions of interest for our present study, although it will break down at earlier times (e.g., Ciardi and Madau 35). With these approximations,

δTb = T0xHI(1 + δρ), (5.2)

1/2 where T0 = 28[(1+ z)/10] mK. Throughout this chapter, we refer to the brightness tem- perature contrast in units of T0.

5.2.2 Semi-Numeric Simulations

In order to simulate reionization we use the ‘semi-numeric’ scheme described in Zahn et al. (189) (see also e.g., Mesinger et al. 119, for related work and extensions to this technique). This scheme is essentially a Monte Carlo implementation of the analytic model of Furlan- etto et al. (62), which is in turn based on the excursion set formalism. The Zahn et al. (189) algorithm allows us to rapidly generate realizations of the ionization field over large simulation volumes at various stages of the reionization process. The results of these calcu- lations agree well with more detailed simulations of reionization on large scales (Zahn et al. 189, 190). We start by generating a realization of the linear density field in a simulation box −1 3 with a co-moving side length of 1 h Gpc and 512 grid cells. The ionization field, xi, is generated following the algorithm of Zahn et al. (189), assuming a minimum host halo mass 8 of Mmin = 10 M⊙, comparable to the atomic cooling mass at these redshifts (Barkana and

Loeb 5). Each halo above Mmin is assumed to host an ionizing source, and the ionizing efficiency of each galaxy is taken to be independent of halo mass. In our fiducial model, we adjust the ionizing efficiency so that the volume-averaged ionization fraction is x = 0.79 at h ii

203 5.2 Method

zfid = 6.9. We focus most of our analysis on this redshift and on this particular model for the volume-averaged ionized fraction. However, we consider additional redshifts in 5.5, as well § as variations around our fiducial ionization history in effort to bracket current uncertainties in the ionization history (see e.g., Kuhlen and Faucher-Giguere 80, Zahn et al. 192). From the linear density field and the ionization field we generate the 21 cm brightness temperature contrast following Equation 5.2. Using the linear density field here – rather than the evolved non-linear density field – should be a good approximation for the large scales of interest for our study; we focus on length scales of R & 20 h−1Mpc and high redshift (z & 6) in subsequent sections.

5.2.3 Redshifted 21 cm Surveys and Thermal Noise

We mostly consider two concrete examples of upcoming redshifted 21 cm surveys. The first is based on the current, 128-tile version of the MWA (Tingay et al. 175) and the second is based on an expanded, 500-tile version of the MWA (as described in Lonsdale et al. 94, and considered in previous work such as Lidz et al. 91, McQuinn et al. 114). These two examples are intended to indicate the general prospects for imaging and bubble identification with first and second generation 21 cm surveys, respectively. Similar considerations would apply for other experiments, but we choose these as a concrete set of examples. We mainly focus on the 500-tile configuration in this chapter because of its greater sensitivity. In 5.5.4, we shift § to considering 128-tile configurations and in 5.6, we consider a LOFAR-style interferometer § for comparison. Hereafter, we refer to the 500-tile configuration as the MWA-500 and the 128-tile version as the MWA-128. Throughout this chapter, we work in co-moving coordinates described by Cartesian la-

bels (x-y-z), with Fourier counterparts (kx-ky-kz). The Fourier modes can be connected directly with the u-v-ν coordinate system generally used to describe interferometric mea- surements. Here u and v describe the physical separation of a pair of antennae in units of the observed wavelength, while ν describes the corresponding observed frequency. The

204 5.2 Method

instrument makes measurements for every frequency, ν, in its bandwidth, and for every antenna tile separation, (u, v), sampled by the array. In order to shift to a Fourier space description, the interferometric measurements must first be Fourier-transformed along the frequency direction. With our Fourier convention, the relation between the two sets of coordinates is given by: 2πu 2πv 2π k = k = k = , (5.3) x D y D z ∆χ where D is the co-moving distance to the survey center and ∆χ is the co-moving distance corresponding to a small difference in observed frequency of ∆ν (e.g., Liu et al. 93). For small ∆ν/ν, we can express ∆χ as c(1 + z ) ∆ν ∆χ fid | |, (5.4) ≈ H(zfid) ν where H(z ) is the Hubble parameter at the fiducial redshift, and ∆ν /ν is the absolute fid | | value of the fractional difference between two nearby observed frequencies. In order to test the prospects for imaging and bubble identification with the MWA, we must corrupt the underlying 21 cm signal described in 5.2.2 with thermal noise. We do § this by generating a Gaussian random noise field in the k-space coordinate system described above, using an appropriate power spectrum. We assume that the co-variance matrix of the thermal noise power is diagonal in k-space. We add the resulting noise field to the underlying 21 cm signal (Equation 5.2). The power spectrum of the thermal noise is given by (McQuinn et al. 113, Furlanetto et al. 59):

T 2 D2∆D λ2 2 P (k,µ)= sys . (5.5) N Bt n(k ) A int ⊥  e  Here µ is the cosine of the angle between wavevector k = k and the line of sight, so | | that k = 1 µ2k is the transverse component of the wavevector. We assume a system ⊥ − 2.3 temperaturep of Tsky = 280 [(1 + z)/7.5] K (Wyithe and Morales 186) and a total observing

time of tint = 1000 hours, which is an optimistic estimate for the observing time in one year.

205 5.2 Method

At our fiducial redshift of zfid = 6.9, the co-moving distance to the center of the survey is D = 6.42 103 h−1Mpc. In this equation, λ denotes the observed wavelength of the × redshifted 21 cm line, λ = 0.211(1 + z)m, and Ae is the effective area of each antenna tile.

We determine Ae by linearly extrapolating or interpolating from the values given in Table 2 2 of Bowman et al. (25); the effective area at zfid = 6.9 is Ae = 11.25m . We assume that the full survey bandwidth of 32 MHz is broken into individual blocks of bandwidth B = 6 MHz to protect against redshift evolution across the analysis bandwidth (McQuinn et al. 114). The co-moving survey depth depth corresponding to a B = 6 MHz chunk −1 is ∆D = 69 h Mpc. The n(k⊥) term describes the configuration of the antenna tiles. More specifically, it is the number density of baselines observing modes with transverse

wavenumber k⊥ (McQuinn et al. 114). Following Bowman et al. (24) and McQuinn et al. (114), we assume the antenna tiles are initially packed as closely as possible in a dense compact core, and that the number density of antenna tiles subsequently falls off as r−2 out to a maximum baseline of 1.5 km. The radius of the dense antenna core is set by the requirement that the antenna density falls off as r−2 outside of the core, and that it

integrates to the total number of antennae. For the MWA-500, this gives rc = 20m, while for the MWA-128, the core radius is r 8m. Equation 5.5 gives the noise power spectrum in c ≈ 2 2 units of mK , and so we divide by T0 to combine with the simulated 21 cm signal expressed

in units of T0. Note that the volume of the MWA survey differs somewhat from that of our reionization simulation. In particular, the transverse dimension of the simulation is smaller than that of the MWA by a factor of 3, while the simulation is deeper in the line-of-sight direction by ∼ about the same factor, as compared with the full MWA bandwidth. However, we remove the long wavelength modes along the line-of-sight direction to mimic foreground cleaning ( 5.2.4), and so we do not, in practice, use the longer line-of-sight scales in our simulation § box. As we will see, the ionized regions in the simulation are substantially smaller than the transverse length of the box. Transverse slices should therefore be representative of what

206 5.2 Method

the actual MWA will observe from a fraction of its larger field of view. We have checked that the coarser transverse k-space sampling in the simulation compared to in the actual MWA survey does not impact our results.

5.2.4 Foregrounds

Next, we need to consider contamination from foreground emission at the frequencies of interest. The relevant foregrounds include diffuse Galactic synchrotron radiation, extra- galactic point sources, and Galactic Bremsstrahlung radiation. While these foregrounds are many orders of magnitude brighter than the cosmological 21 cm signal, they are expected to individually follow smooth power laws in frequency. Over a sufficiently small frequency range, the summed contributions can also be approximated as following a smooth power law, while the 21 cm signal will vary rapidly. This allows the foregrounds to be removed from the data by, for example, fitting a low-order function along each line of sight and subtracting it. While this procedure is effective at removing foreground contamination, it also removes long wavelength modes along the line of sight from the signal itself, and hence prevents measuring these modes. Several related methods for foreground removal have been discussed in the literature (e.g., Wang et al. 180, Harker et al. 67, Petrovic and Oh 143, Chapman et al. 32). In this work, we approximately mimic the degrading effects from fore- ground removal by subtracting the running mean from the noisy signal along each line of sight, rather than including realizations of the foregrounds in our simulation and excising them with one of the above algorithms. We generally remove the running mean over a −1 bandwidth of 16 MHz, which corresponds to a co-moving distance of Lfg = 185 h Mpc at redshift z = 6.9; we consider the impact of other choices of L in 5.5.3. We defer more fid fg § detailed models of foreground contamination, and foreground removal algorithms, to future work.

207 5.3 Prospects for Imaging

5.3 Prospects for Imaging

Having described our mock 21 cm data sets, we now turn to consider the prospects for constructing direct ‘images’ of the redshifted 21 cm signal. Previous studies already suggest that the prospects for imaging with the MWA-500 are limited (McQuinn et al. 114). Here we emphasize that even a crude, low-resolution image of the redshifted 21 cm signal may be quite interesting, especially given that the ionized regions during reionization may be rather large scale features. We hence seek to quantify the imaging capabilities further using our corrupted reionization simulations. Here our work complements recent work in a similar vein by Zaroubi et al. (196), who considered the prospects for imaging with LOFAR. While the central idea in this section is similar to this previous work, we focus on the MWA while Zaroubi et al. (196) considered LOFAR. In order to construct the best possible images from the noisy mock 21 cm data, we apply a Wiener filter. We assess the ability of the MWA to image the redshifted 21 cm sky by comparing the filtered (recovered) noisy signal with the underlying noise-free 21 cm input signal.

5.3.1 The Wiener Filter

The Wiener filter is the optimal filter for extracting an input signal of known power spectrum when it is corrupted by additive noise, also with known power spectrum. As described in Press et al. (146), this filter is optimal in that it minimizes the expectation value of the integrated squared error between the estimated signal field and the true signal field. The estimate of the true signal is a convolution of the Wiener filter and the corrupted signal in real space, and so is a product of the two quantities in Fourier space,

S˜(k)= C(k)W (k), (5.6)

where C(k), W (k), and S˜(k) are the Fourier transforms of the corrupted signal, Wiener filter, and estimated signal, respectively. Requiring that the filter be optimal in the least-

208 5.3 Prospects for Imaging

square sense results in W (k) taking the form

P (k) W (k,µ)= S , (5.7) PS(k)+ PN(k,µ) where PS(k) and PN(k,µ) are the power spectra of the signal and noise, respectively. We note that, while the signal power spectrum is roughly isotropic1, the noise power spectrum depends on µ and consequently so does the filter. The filter keeps a unity weighting for k- modes where P (k) P (k) and significantly downweights k-modes where P (k) P (k). S ≫ N S ≪ N This can allow for partial recovery of the original signal, provided that the signal power dominates for some k-modes.

The Wiener filter requires an estimate of the signal power spectrum, PS(k), and of the total (signal plus noise) power spectrum, PS(k)+ PN(k,µ), as inputs. These may not be precisely known. However, since the filter is the outcome of a minimization problem – i.e., it minimizes the expected difference between the estimated and true fields – the accuracy of the filter should be insensitive to small changes about its optimal value. In other words, the accuracy of the filter is not expected to change greatly by using estimates of the signal and noise power spectra rather than the true spectra. Furthermore, we do expect to have an estimate of the underlying signal power spec- trum; measuring this statistic is a major goal of redshifted 21 cm surveys. Specifically, the underlying signal power can be estimated by cross-correlating redshifted 21 cm measure- ments made over two different time intervals (after foregrounds have been removed). The statistical properties of the signal should be identical across the two different time periods, but the thermal noise contributions will be independent. The cross-correlation between two time chunks then provides an unbiased estimate of the signal power (e.g., Liu et al. 93). Es- timates of the noise power spectrum can then be made by subtracting the estimated signal

1Redshift-space distortions and redshift evolution across the observed bandwidth break isotropy (e.g. Datta et al. 43). However, for the bandwidth considered here (B = 6 MHz) and the neutral fractions of interest, the signal should be approximately isotropic.

209 5.3 Prospects for Imaging

power from the power measured over the entire integration time, which contains both the signal and noise contributions. The Wiener filter does not actually require the noise power spectrum to be known on its own. However, in 5.4 we consider the optimal matched filter, § which does have this requirement. Throughout this study, we assume perfect knowledge of the underlying power spectra. Before applying the Wiener filter to our corrupted simulations, it is useful to estimate the expected signal-to-noise ratio of the filtered maps analytically, using simulated signal power spectra and the noise power spectrum of Equation 5.5. The expected signal-to-noise S 2 ratio of the Wiener-filtered field is wf =σ ˜S/σ˜N , whereσ ˜S(N) is the filtered signal (noise) variance. The signal and noise variance can in turn be calculated as integrals over their respective power spectra, d3k σ˜2 = W (k,µ) 2P (k,µ). (5.8) S(N) (2π)3 | | S(N) Z Here we use PS(N) to denote the power spectrum of the signal (noise). One can also consider the impact of foreground cleaning here by downweighting modes where the foreground power is large compared to the signal power. In order to consider the dependence of the signal- to-noise ratio on the stage of reionization, we consider simulation outputs in which the volume-averaged ionization fraction is x = 0.51, 0.68, 0.79 and 0.89. We consider each of h ii 1 these models at our fiducial redshift of zfid = 6.9. Presently, we don’t consider still earlier stages of reionization since the prospects for imaging with the MWA-500 are especially poor for lower ionized fractions. The resulting Wiener filters for the different values of x are shown in Figure 5.1, after h ii integrating over angle µ. In this figure, foreground cleaning has been accounted for by

1In practice, the simulated ionization fields for ionized fractions lower (higher) than our fiducial value

(hxii = 0.79 at zfid = 6.9) come from slightly higher (lower) redshift simulation outputs. We generate the

21 cm signal and noise as though each data cube were in fact at zfid = 6.9. This is appropriate to the extent that the statistical properties of the ionized regions are mainly determined by the volume-averaged ionized fraction, and are relatively insensitive to the precise redshift at which a given ionized fraction is reached (see McQuinn et al. 110 and Furlanetto et al. 62.)

210 5.3 Prospects for Imaging

0.9 = 0.51 i 0.8 = 0.68 i = 0.79 0.7 i = 0.89 i 0.6

0.5

W(k) 0.4

0.3

0.2

0.1

0 −1 0 10 10 k (h−1Mpc)−1

Figure 5.1: Fourier profile of the Wiener filter, W (k). The filter is averaged over line-of-sight

angle and the results are shown at zfid = 6.9 for simulated models with x = 0.51 (blue h ii dotted), x =0.68 (cyan dot-dashed), x =0.79 (green dashed), and x =0.89 (red solid). h ii h ii h ii subtracting a running mean along the line of sight, as described in 5.2.4. It is helpful to § note, from Equation 5.7, that the filter is equal to 1/2 for modes where the signal and noise power are equal. The figure suggests that a small range of k-modes with k . 0.1h Mpc−1 will have signal-to-noise ratio larger than unity for all four ionized fractions considered, although imaging is less promising for the smaller ionized fractions. If the ionized regions are larger than in our fiducial model – as expected if, for example, rarer yet more efficient and more clustered sources dominate reionization (e.g., McQuinn et al. 110, Lidz et al. 91) – then the prospects for imaging may improve somewhat. Performing the integrals in Equation 5.8, while incorporating foreground cleaning, we find that the total signal-to-noise ratio expected for the MWA-500 is S = 0.52, 0.79, 1, and 1.2 for x = 0.51, 0.68, 0.79, 0.89, wf h ii respectively.

211 5.3 Prospects for Imaging

5.3.2 Application to a Simulated 21 cm Signal

With the analytic signal-to-noise ratio estimates as a guide, we apply the Wiener filter to our mock noisy redshifted 21 cm data. The results of these calculations, for a particular slice through the simulation volume, are shown in Figure 5.2. The side length (1 h−1Gpc) of each slice is a factor of 3 smaller than the transverse dimension of the MWA. One ∼ can asses how well the original signal is ‘recovered’ by comparing the top-left panel of the figure which shows the input signal with the bottom-left panel which shows the filtered noisy signal, after mimicking foreground removal. The two panels do not bear a striking resemblance since the average signal-to-noise ratio is only of order unity. Nonetheless, it is encouraging that many of the minima in the filtered noisy signal do indeed correspond to ionized regions in the input signal. Furthermore, we can compare the filtered noisy signal in the bottom-left panel with the top-right panel, which shows filtered pure noise. While these two panels do not look drastically different, they are easily distinguishable from each other given the increased contrast in the filtered noisy signal. In addition, we see that the filtered noisy signal obtains signal-to-noise values exceeding 6 σ, while the statistical − significance of the filtered noise does not exceed 5 σ. Quantitatively, 3% ( 0.03%) ∼ − ∼ ∼ of the volume in the filtered noisy signal is occupied by pixels with statistical significance greater (in magnitude) than 3 σ (5 σ). This is expected given that the filtered data cube − − has an average signal-to-noise ratio of σ /σ 1, as anticipated in the analytic calculation S N ≈ of 5.3.1. § Comparing the filtered noisy signal and the filtered pure noise, one can see that ionized regions in the underlying signal are diminished if they happen to be coincident with upward fluctuations in the noise, as expected. For example, the ionized region in the bottom-right corner of the unfiltered signal lies very close to a 3 σ upward fluctuation in the filtered ∼ − noise and, as a result, appears with weak statistical significance in the filtered noisy signal. Conversely, some of the most statistically significant regions in the filtered noisy signal occur when ionized regions overlap downward noise fluctuations. We can further compare

212 5.4 Prospects for Identifying Ionized Regions

the filtered noisy signal with the filtered noise-less signal, shown in the bottom right panel of Figure 5.2. The filtered noise-less signal is normalized by the standard deviation of the filtered noise so that it can be compared with the signal-to-noise slices in the other panels. This comparison reveals that high significance regions (& 5 σ) in the filtered − noisy signal only line up well with the corresponding regions in the filtered noiseless signal if they are coincident with downward fluctuations in the noise. On its own, the filtered

noiseless signal only attains statistical significances of . 4σN. Finally, Figure 5.3 illustrates the impact of foreground cleaning, performed here over a bandwidth of 16 MHz ( 5.2.4). § Foreground cleaning removes the long wavelength modes along the line of sight – which is along the vertical axis in the figure – and thereby compresses structures along the line of sight. However, the cleaning process only impacts the long wavelength line-of-sight modes which still leaves room to image other modes robustly. Note that the slice thickness (8 h−1Mpc) in Figure 5.2 and 5.3 is somewhat arbitrary. However, the Wiener filter smooths out structure on significantly larger scales than this (Figure 5.1), and so we expect similar results for other values of the slice thickness, provided the slice is thin compared to the cut-off scale of the filter. In practice, of course, one can make many independent maps similar to Figure 5.2 from the MWA-500 or similar surveys. Collectively, our results mostly confirm previous wisdom; the prospects for imaging with the MWA-500 are limited. Nonetheless, it appears that a signal-to-noise ratio of order unity is achievable, suggesting that the MWA-500 can make low resolution maps of the reionization process.

5.4 Prospects for Identifying Ionized Regions

We now shift our focus to discuss whether it may also be possible to identify interesting individual features in upcoming 21 cm data cubes. In particular, we aim to identify ion- ized regions in noisy 21 cm data sets and, furthermore, to estimate the spatial center and approximate size of each ionized bubble. For this purpose, we will use an optimal matched

213 5.4 Prospects for Identifying Ionized Regions

filter technique. As we discuss, individual ionized regions may be identifiable as prominent minima in the filtered field.

5.4.1 The Optimal Matched Filter

The optimal matched filter is suited for the case of a corrupted signal containing a known feature that one would like to extract. The filter acts in Fourier space by cross-correlating the corrupted signal with a template describing the known feature, while downweighting k-modes in the corrupted signal by the noise power. The resulting form of the filter in Fourier space, M(k,µ), is T (k) M(k,µ)= , (5.9) PN(k,µ) where T (k) is the Fourier profile of the known feature. The filter is optimal in the sense that it maximizes the signal-to-noise ratio in the filtered data cube at the location of the feature being extracted. While the Wiener filter requires an estimate of the signal and total (signal plus noise) power spectra, the matched filter requires a good estimate of the template profile, T (k), and the noise power spectrum, PN (k,µ). For our present application, we would like templates describing the ionized regions. An appropriate choice is not obvious; theoretical models predict that the ionization state of the gas during reionization has a complex, and somewhat uncertain, morphology, with ionized regions of a range of sizes and shapes (Iliev et al. 76, Zahn et al. 189, McQuinn et al. 110). However, we find that the simplest conceivable choice of template filters, corresponding to completely ionized spherical bubbles of varying size, are nonetheless effective at identifying ionized regions with a more realistic and complex morphology. In this case, T (k; R) is just the Fourier transform of a spherical top-hat of radius R and is given by V k T ( ; RT)= 3 3 [ kRT cos kRT + sin kRT] , (5.10) k RT −

214 5.4 Prospects for Identifying Ionized Regions

with V denoting the volume of the spherical top-hat. Note that the precise normalization of the filter is unimportant since we are mainly interested in the signal-to-noise ratio here, in which case the overall normalization divides out.

5.4.2 Application to Isolated Spherical Ionized Regions with Noise

It is instructive to first consider an idealized test case that can be treated analytically before applying the matched filter to our full mock 21 cm data sets. In particular, we consider the case of an isolated, spherical, and highly ionized region placed at the origin and embedded in realistic noise. We assume that the neutral fraction exterior to the ionized region is uniform, with a mass-weighted neutral fraction of x (1 + δ ) . Ignoring foreground contamination h HI ρ i for the moment, the 21 cm signal may be written as:

δT (x) δT = B˜(x; R )+ N˜ (x), (5.11) b − h bi B where B(x; RB) denotes our isolated bubble of radius RB, and N(x) denotes the thermal noise contribution to the signal. We have subtracted off the overall mean brightness tem- perature, δT , since this will not be measured in an interferometric observation. The h bi tildes indicate that the spatial average has been removed from each of the underlying signal and noise so that B˜(x; RB) and N˜ (x) each have zero mean. In this case B˜(x; RB) has an inverted spherical top-hat profile,

xHI(1 + δρ) x < RB, B˜(x; RB)= − h i | | (5.12) (0 otherwise.

The Fourier transform of the isolated bubble is hence related to the Fourier transform of our template by B˜(k; R ) = x (1 + δ ) T (k; R ). Note that we express brightness B − h HI ρ i B temperatures in units of T0 (see Equation 5.2), and so all quantities here are dimensionless. It is straightforward to derive the expected signal-to-noise ratio at the center of the isolated ionized region, and thereby gauge the prospects for bubble detection with a matched

215 5.4 Prospects for Identifying Ionized Regions

filter technique. Let us assume that the radius, RB, of our template filter is well matched to the true radius of the ionized region. This will maximize the expected signal-to-noise ratio. Neglecting foregrounds for the moment, and using the fact that the thermal noise has zero mean, we find that the signal-to-noise ratio at bubble center for the optimal matched filter is: 1/2 d3k T 2(k; R ) S(R )= x (1 + δ ) B . (5.13) B − h HI ρ i (2π)3 P (k,µ) Z N  For our sign convention, in which the template and ionized regions have opposite signs, this quantity is negative – ionized bubbles are regions of low 21 cm signal. The contribution 2 of a Fourier mode to the signal to noise ratio depends on the relative size of T (k; RB) and PN(k,µ): modes for which the template is much larger than the noise power contribute

appreciably to S(RB) while modes dominated by the noise power are not useful. The signal- to-noise ratio depends on the neutral fraction: a larger exterior neutral fraction increases the contrast between an ionized bubble and the exterior, and hence boosts the detectability of the ionized region. We would like to calculate the expected signal-to-noise ratio for ionized regions of different sizes and for various volume-averaged ionization fractions. To do this, we need to connect the volume-averaged ionized fraction with the mass-averaged fraction, x (1 + δ ) , which enters into Equation 5.13. Here we should incorporate that large scale h HI ρ i overdense regions are generally ionized before typical regions during reionization, i.e., the neutral fraction and overdensity fields are anti-correlated. Defining δ = (x x )/ x , x HI−h HIi h HIi we approximate δ δ as fixed at δ δ = 0.25 throughout the reionization process (Lidz h x ρi h x ρi − et al. 89). The results of the signal to noise calculation are shown in Figure 5.4 for the MWA-500

and a LOFAR-type experiment. Here we consider only our fiducial redshift, zfid = 6.9. The

(absolute value of) S(RB) is evidently a strongly increasing function of bubble size. This occurs because the thermal noise is a strong function of scale and only the rather large scale modes are measurable. It is encouraging that the expected signal-to-noise ratio exceeds five, S(R ) & 5, for a range of radii and neutral fractions. This corresponds to a 5 σ B −

216 5.4 Prospects for Identifying Ionized Regions

detection: ‘false’ bubbles at this significance from downward fluctuations in the noise are highly unlikely, with a fraction of only 3 10−7 of pixels in the filtered noise having ∼ × such a large (negative) significance on their own. For simplicity we neglect the impact of foreground cleaning in this figure: this will degrade the expected signal-to-noise ratios somewhat, as we will consider subsequently (see 5.4.3, 5.5.3). § § In order to estimate the number of bubbles that can be detected from these curves, we need to consider how many bubbles there are of different sizes, i.e., we need to fold in an estimate of the bubble size distribution. In particular, while the contrast of an ionized region increases with the neutral fraction, large ionized bubbles become increasingly scarce for larger values of the neutral fraction. For instance, we can consider the model bubble size distributions in Figure 4 of Zahn et al. (189). This figure indicates that bubbles of radius larger than 30 h−1Mpc are exceedingly rare for neutral fractions larger than x > 0.5, h HIi with only the tail end of the distribution extending past 25 h−1Mpc. However, bubbles this size are relatively common later in reionization. Since Figure 5.4 indicates that only −1 bubbles with R & 30 h Mpc exceed S(RB) & 5, this suggests that bubble detection is feasible for the MWA-500 after the Universe is more than 50% ionized, but that it will ∼ be difficult to use this method at earlier stages of the reionization process. Also, note again that the the calculation here neglects the effects of foreground cleaning. However, we find that incorporating foreground cleaning only has a small effect on bubbles of this size (. 30 h−1 Mpc , 5.5.3). Bubble detection will also be challenging once the Universe is less § than 10 20% neutral, owing mostly to the reduced contrast between the bubbles and ∼ − typical regions. If the ionized bubbles at a given stage of the EoR are larger than in the model of Zahn et al. (189), then the prospects for bubble detection will be enhanced. We refer the reader to McQuinn et al. (110) for a quantitative exploration of the bubble size distribution across plausible models for the ionizing sources. Finally, it is interesting to consider a LOFAR-style interferometer, as discussed further in 5.6. This is shown as the red dot-dashed curve in Figure 5.4. The expected S(R) exceeds §

217 5.4 Prospects for Identifying Ionized Regions

that of the MWA-500 for small bubble radii, before flattening off at larger radii. This occurs because the LOFAR-style interferometer has more collecting area per baseline, but a larger minimum baseline. This makes it more sensitive to the smaller ionized regions, but less sensitive to larger ones. While the signal-to-noise curves in this toy case provide a useful guide, we should keep in mind their limitations. First, it considers only the case of a single isolated ionized region. Next, we consider here only the signal to noise at the bubble center, while an ionized region will typically have a strong (negative) signal to noise over much of its volume. This can help significantly with detection. Finally, we consider only the average signal-to-noise ratio here. In practice, the signal-to-noise ratio in a filtered map may fluctuate significantly around this average, as we will see.

5.4.3 Application to a Simulated 21 cm Signal

With the estimates of the previous section as a rough guide, we now apply the matched filter to our noisy mock redshifted 21 cm data. In order to illustrate the results of passing our mock data through a matched filter, we start by examining simulated signal-to-noise fields −1 for a single template radius of RT = 35 h Mpc. This template radius corresponds to the typical size of the ionized bubbles we believe we can detect (see Figure 5.4). A representative slice through the simulation is shown in Figure 5.5. The results look promising, with signal- to-noise ratios comparable to the values anticipated in the idealized calculation of Figure 5.4. Although the Wiener filter provides the best overall map, or data cube, one can still detect individual features at greater significance by applying a matched filter. Comparing with Figure 5.2, it is clear that the Wiener filter is passing more small scale structure than the matched filter shown here. This results in the signal-to-noise ratio being larger (in absolute value) for the matched filter than for the Wiener filter. In particular, we find values of the signal-to-noise ratio that are as low as 10 in the matched filter data cube, a significant ∼− improvement over the global minimum of 6 for the Wiener filter. Moreover, we can ∼ −

218 5.4 Prospects for Identifying Ionized Regions

compare the filtered noisy signal in the bottom-left panel with the filtered pure noise field in the top-right panel. They differ by more than in the case of the Wiener filter. Indeed, the very low signal to noise ratio regions (shown in dark blue/purple in the bottom-left panel) line up fairly well with ionized regions in the top-left panel. This is especially apparent when comparing the filtered noisy signal to the filtered noiseless signal, shown in the bottom-right panel. For the slice shown, almost all of the significant features in the filtered noiseless signal are preserved in the noisy case. Figure 5.6 shows the impact of foreground cleaning: as in the case of the Wiener filter (Figure 5.3), this compresses structures along the line of sight and reduces the overall signal-to-noise ratio in the data cube. The signal-to-noise ratio is still significant enough, however, to robustly identify ionized regions. As with the Wiener filter, and in what follows subsequently, we show slices of 8 h−1Mpc thickness. This choice is arbitrary, but we expect similar results provided the slice thickness is small compared to the radius of the template filter. It is important to keep in mind, however, the full data cube will consist of many separate slices of this thickness. Also note that the transverse dimension of the MWA-500 is larger than that of our simulation box by a factor of 3, and so these slices represent only 1/9 of the MWA field of view. ∼ ∼ These results are promising, but they are for a single filtering scale, and so we can do significantly better by considering a range of template radii, and looking for extrema in the resulting signal-to-noise fields. In particular, we proceed to apply a sequence of filters with template radii up to R 75 h−1 Mpc – see 5.4.5 for a justification of this maximum T ≤ § – across the simulation volume. We assign the minimum (most negative) signal-to-noise value obtained over the range of template radii to each simulation pixel and use this to construct a new field. The position of local minima in this field are chosen to be the centers of candidate bubbles, and each such bubble is assigned a radius according to the scale of the template filter that minimizes its signal-to-noise. We focus on minimum values since ionized regions are expected to appear as regions of low 21 cm signal. All candidate bubbles whose central signal to noise is lower than 5 are considered to be detected ionized regions. −

219 5.4 Prospects for Identifying Ionized Regions

We find it important to apply one additional criterion to robustly identify ionized re-

gions. The criterion is that a low signal to noise region on scale RB must additionally be

low in signal to noise at all smaller smoothing scales, RT < RB. This guards against the possibility that a detected bubble will be centered on neutral material that is nevertheless surrounded by ionized hydrogen. A region like this will have a high (least negative) signal- to-noise when filtered on small scales and then dive down (gaining statistical significance) when filtered on scales containing the surrounding ionized material. We discard such spu- rious bubbles by requiring that the field is low on all smaller smoothing scales. The only downside to this procedure is that it occasionally discards true ionized regions whose center happens to coincide with a significant upward noise fluctuation. Overall, however, it im- proves the quality of detected bubbles ( 5.4.4). This cut also requires a threshold choice; we § reject candidate regions if their signal-to-noise ratio crosses above a threshold Smax at any smoothing scale less than RT. After trying several thresholds, we found the most effective choice to be S = 1σ. In principle, one might use the full curve of signal-to-noise ratio max − versus template radius for each candidate bubble to help verify the detection and determine the properties of the bubble. In practice, we found that individual signal-to-noise curves are noisy and difficult to incorporate into our analysis and so we don’t consider this possibility further in what follows. We apply this algorithm to the mock redshifted 21 cm data and identify 220 ionized regions across the simulation volume (which is different than the MWA survey volume, as we will discuss subsequently). A representative example of a detected bubble is shown in Figure 5.7. The circle in the figure identifies the detected bubble size and the location of its center in both the filtered noisy signal (top-left and top-right panels), as well as in the input signal (bottom-left and bottom-right panels). The algorithm has convincingly identified an ionized region. The detected bubble overlaps a small fraction (%10) of neutral material in the input signal. Although this particular ionized bubble is well identified, most of the ionized regions in the signal will escape detection. This is because the significance levels of the detected

220 5.4 Prospects for Identifying Ionized Regions

bubbles are not that high, and an ionized region generally needs to be coincident with a downward fluctuation in the noise to pass our significance threshold. For example, consider the larger ionized region below and to the left of the detected region in the bottom-left panel of Figure 5.7. This region, while larger and therefore more detectable on average than the identified bubble, happens to coincide with a large upward noise fluctuation and hence fails to cross the significance threshold. While we can not identify all of the large ionized regions in the noisy mock data, we can robustly identify some regions; this may still be quite valuable. It is also clear that the underlying ionized regions are manifestly non-spherical, creating an ambiguity as to what the appropriate ‘radius’ of the region is. Focusing on the bot- tom right panel in Figure 5.7, we could imagine the size being reasonably described by a radius %50 larger, so as to enclose more of the nearby ionized material. However, our ∼ method naturally favors radii causing little overlap with neutral material at these size scales. Therefore, an ionized region like the one shown in Figure 5.7 is more likely to be detected as several small ionized regions than one large one, although both characterizations seem reasonable. Figure 5.8 gives a further example of how the algorithm identifies bubbles, and some of the ambiguities that can result. This figure shows an example of an irregular, yet contiguous, ionized region that is detected as more than one ionized bubble. Here we show spatial slices through the center of the middle sphere, marked with a solid circle, which happens to intersect neighboring ionized bubbles, whose cross sections are shown as dashed circles. Hence, our algorithm generally represents large, irregularly shaped, yet contiguous, regions as multiple ionized bubbles. It is important to emphasize further the difference between the simulated results shown here and the idealized test case of the isolated bubble shown in the previous section. In particular, we consider here the application of matched filters to the 21 cm signal during a late phase of reionization in which many ionized regions, with a broad size distribution, fill

221 5.4 Prospects for Identifying Ionized Regions

the survey volume: the ionized regions are not isolated bubbles in a sea of partly neutral material. When applying a matched filter of template radius RT around a point, the values of the field at many neighboring pixels impact the filtered field at the point in question. It is hence possible that a filtered pixel is affected by several distinct neighboring ionized regions. Indeed, this can result in even neutral regions having low signal-to-noise ratios provided they are surrounded by many nearby ionized regions. For instance, in the low noise limit, any region with volume-averaged neutral fraction lower than the cosmic mean would pass our significance threshold. To guard against this type of false detection, we implemented the requirement that a candidate bubble has low signal-to-noise for all template radii smaller than the detected radius. Another possibility might be to treat small ionized regions as an additional noise term in the filter. However, in practice, our attempts along these lines introduced an additional level of model dependence without significantly increasing the quality of the detected bubbles. Ultimately, it is important to keep in mind that the signal- to-noise values quoted here reflect only the likelihood that a value arises purely from noise, and so they are not strictly indicative of the quality of the detected bubbles.

5.4.4 Success of Detecting Ionized Regions

We hence turn to describe the characteristics of the detected ionized regions, and to quantify the method’s level of success in detecting ionized bubbles. To do this, we calculate the fractional overlap of each detected bubble with ionized material in the underlying signal. Additionally, we estimate how many ionized bubbles should be detectable across the entire MWA-500 survey volume. The matched filter technique finds 220 bubbles across our simulation volume. However, the algorithm for determining bubble positions and sizes allows for bubbles to occupy over- lapping areas, as shown in Figure 5.8. We find that 55% of the detected bubbles have some ∼ overlap with another bubble, although only 15% of the total volume occupied by detected ∼ bubbles is occupied by more than one. Regardless, 96% of the detected ionized bubbles ∼

222 5.4 Prospects for Identifying Ionized Regions

have an average ionized fraction larger than xi = 0.79, which is the volume-averaged ioniza- tion fraction of the simulation box at the redshift under consideration. Furthermore, 42% ∼ of the detected bubbles have an ionized fraction greater than xi = 0.9. The lowest ioniza- tion fraction of a detected bubble is xi,min = 0.77, just slightly below the volume-averaged ionization fraction of the simulation. In total, we detect 9 bubbles whose ionized fractions are less than the average ionization fraction of the box. Inspection reveals that these regions happen to be coincident with significant ( 3 σ) downward noise fluctuations. ≤− − In Figure 5.9, we plot the volume-averaged ionized fraction within each of our detected bubbles against the detected bubble radius. For comparison, we show the 1 σ spread in − the ionized fraction enclosed by randomly distributed spheres of the same size.1 The spread in ionization of the randomly distributed spheres around the box average ionized fraction, x = 0.79, decreases with increasing radius; this reflects the drop off in the power spectrum h ii of the ionization field towards large scales. Most of the detected bubbles are significantly more ionized than random regions, as expected, indicating a significant success level. There are a few poor detections which result mostly from downward noise fluctuations. There is a small overall decrease in the ionized fraction of detected bubbles larger than RB & 40 h−1Mpc, suggesting that we may no longer be detecting individual ionized regions here. These regions may potentially be distinguished from isolated bubbles by examining the signal-to-noise ratio as a function of template radius closely, as we discuss in 5.4.5. § We can estimate the number of ionized regions detectable in the MWA-500 by scaling from our simulation volume to the MWA survey volume. At zfid = 6.9, for an ionized fraction of x = 0.79, we expect to find 140 bubbles in a B = 6 MHz chunk of the MWA, over its h ii entire field of view of 770 deg2. About 135 (60) of these detected bubbles are expected ∼ to have ionized fractions larger than 79% (90%). This estimate comes from simply scaling our simulation volume (which is deeper than the MWA bandwidth) to a 6 MHz portion

1 The 1 − σ spread shown in the figure extends past xi = 1, but this is only because the distribution of ionized fractions is not symmetric about the mean, i.e., the probability distribution function of the ionized fraction is non-Gaussian.

223 5.4 Prospects for Identifying Ionized Regions

of the MWA survey volume. Analyzing the MWA data over a 6 MHz chunk is meant to guard against redshift evolution: the full bandwidth of the survey is B = 32 MHz and so the prospects for bubble detection across the full survey are even better than this estimate suggests. The precise gain will be dependent on how rapidly the bubble size distribution evolves across the full survey bandwidth. One caveat with our estimate, however, is that B = 6 MHz corresponds to only 70 h−1 Mpc . This is comparable to the size of our ∼ larger bubbles, and so analyzing chunks this small might weaken our ability to detect large bubbles. This effect is not incorporated in our scaling estimate, which simply takes the ratio of the MWA survey volume and our simulation volume. In practice, one can perform the bubble extraction for different analysis bandwidths to help ensure robust detections.

5.4.5 Range of Template Radius Considered

It is worth mentioning one further detail of our algorithm. In the previous section, we set −1 the maximum template radius considered at RT,max = 75 h Mpc, without justification. In fact, we have a sensible and automated way for arriving at this choice. We discuss this procedure briefly here. A good candidate ionized region should in fact obey three criteria. First, it should have a large (negative) signal-to-noise ratio, so that it is unlikely to result from a noise fluctuation. Second, the signal-to-noise ratio should be small for all trial radii smaller than the optimal template radius, as discussed in 5.4.3. Finally, the total signal must itself be small in an § absolute sense. In the limit of low noise, anything less neutral than average would qualify as a bubble by the first criterion, and so this third criterion may then become important for robustly identifying bubbles. This low noise limit is relevant for the MWA-500 only on very large smoothing scales, where the noise averages down significantly. Since this third criterion becomes important only on very large smoothing scales here, we use it only to set the maximum template radius considered. Without this third con- sideration, our algorithm generally identifies a few excessively large ionized bubbles, but

224 5.4 Prospects for Identifying Ionized Regions

this can be easily understood and avoided as follows. Consider, for the moment, the 21 cm brightness temperature field in the absence of noise and foregrounds. Let’s further work in units of T0 (Equation 5.2), and remove the average brightness temperature contrast across the data cube. In this case, the signal inside a highly ionized bubble is expected to be x (1 + δ ) . If we now spherically average the field on scales smaller than the bubble, −h HI ρ i the value at bubble center will not change from this value, x (1+δ ) . Once the smooth- −h HI ρ i ing scale becomes larger than the bubble scale, however, surrounding neutral material will increase the value of the filtered field at bubble center. Hence, if the filtered field becomes everywhere larger than x (1 + δ ) on some smoothing scale, it is clear that no larger −h HI ρ i ionized bubbles exist within the data cube. This suggests that we can set the maximum template radius by requiring that the filtered noisy signal reaches sufficiently small values, at some locations across the data cube, for there to still plausibly be completely ionized regions. Since the presence of noise only increases the variance, this should provide a con- servative estimate of the maximum size of the ionized regions. In practice, we need to chose a threshold criterion without assuming prior knowledge of the neutral fraction. Here we set the maximum template radius to be the smallest smoothing scale at which the filtered noisy field everywhere exceeds x (1 + δ ) 0.075. This corresponds to the expected −h HI ρ i≥− contrast at x = 0.1, assuming δ δ = 0.25, and yields a maximum template radius h HIi h x ρi − of R = 70 h−1 Mpc , 73 h−1 Mpc , 75 h−1 Mpc , and 75 h−1 Mpc for x = 0.51, 0.68, T,max h ii 0.79, and 0.89, respectively. The precise threshold used here, 0.075, is somewhat arbitrary − but this choice is only being used to set the maximum template radius considered.1

1This choice might appear to preclude the possibility of detecting bubbles at the end of reionization

when hxHI(1+ δ)i≤ 0.075. However, the threshold choice is only used to set the maximum template radius, and so ionized regions may still in principle be detected at these late stages of reionization. The ionized regions identified at the end of reionization are, however, generally less robust given the reduced contrast between fully ionized and average regions at these times (see §5.5.1).

225 5.5 Variations on the Fiducial Model

5.5 Variations on the Fiducial Model

So far, we have considered the prospects for bubble detection only in our fiducial model with x = 0.79 at z = 6.9 and only for the MWA-500. Here we consider first alternate models h ii fid in which the Universe is more or less ionized at zfid = 6.9 than in our fiducial case, and then consider how the sensitivity declines towards higher redshifts at fixed ionized fraction. In addition, we consider variations around our fiducial assumptions regarding the impact of foreground cleaning. Then we turn to consider the sensitivity of the MWA-128; this is meant to illustrate the prospects for bubble detection with the very first generation of redshifted 21 cm surveys, while the MWA-500 represents a second generation survey.

5.5.1 Ionized Fraction

In order to consider bubble finding at earlier and later stages of the EoR, we run our matched filter on simulation outputs with volume-averaged ionized fractions of x = 0.51, 0.68, and h ii 0.89. As discussed in 5.3.1 these outputs are actually at slightly different redshifts, but we § generate the 21 cm field as though they were at zfid = 6.9. As far as bubble detection is concerned, varying the ionized fraction leads to two, generally competing, effects. First, the bubbles grow as reionization proceeds. This tends to boost detection, since it is only the large scale modes that are detectable over the thermal noise. Second, however, the contrast between an ionized region and a typical volume of the Universe is reduced as reionization proceeds. This makes bubble detection more difficult. Both of these effects are quantified in the idealized isolated bubble case in Figure 5.4. It is also clear that the ideal ionized fraction for bubble detection will be somewhat survey dependent. As already illustrated in Figure 5.4 and discussed further in 5.6, a LOFAR-type interferometer will perform better § when the ionized regions are smaller. We find that the matched filter is capable of detecting ionized regions for each of the ionized fractions studied. In Figure 5.10 we show histograms of the detected bub-

226 5.5 Variations on the Fiducial Model

ble size distributions for each ionized fraction. Since we preferentially detect large ionized regions, we don’t expect these distributions to be representative of the true underlying bubble size distributions. For example, in Figure 5.10, the size distribution peaks around −1 & 40 h Mpc for the case of xi = 0.79, despite volume-weighted size distribution peaking around 30 h−1 Mpc in Figure 4 of Zahn et al. (189) at roughly the same ionized fraction. ∼ Nonetheless, the histograms illustrate a general shift from smaller to larger detected bubble radius as the ionized fraction increases. By applying the matched filter to several redshift bins, one can potentially observe precisely this evolution with the MWA-500. This would complement studies of the 21 cm power spectrum evolution over the same redshift range (e.g., Lidz et al. 91). From the histograms, one can see that – of the models shown – the best ionized fraction for bubble detection is x = 0.79. This is apparently near the sweet h ii spot for the MWA-500 where the bubbles are large enough in the model for detection, but the contrast with typical regions is still sufficiently large. The average ionized fraction within detected bubbles varies significantly across the dif- ferent ionized fractions considered. Specifically, the percentage of detected bubbles that are more than 90% ionized is 0%, 15%, 43%, and 91% for x = 0.51, 0.68, 0.79, and h ii 0.89, respectively. However, in each case the percentage of detected bubbles with ionized fraction larger than the (global) volume-averaged ionization fraction is fixed at & 95%. At first glance, one aspect of these results may appear to be in tension with the calculations of 5.4.2, where we estimated that bubble detection would be unsuccessful for neutral fractions § larger than x & 0.5. However, this estimate considered the detection of isolated bub- h HIi bles. Inspection reveals that the detected bubbles at x = 0.51 each correspond to clusters h ii of smaller ionized regions. Evidently, these appear as a single larger ionized region after convolving with the template filter and downweighting the noisy short-wavelength modes. In practice, it may be possible to distinguish this case from that of an isolated bubble by analyzing the trend of signal-to-noise ratio versus trial template radius. The signal-to-noise

227 5.5 Variations on the Fiducial Model

ratio is expected to grow more rapidly with radius (before reaching the bubble scale) for a truly isolated bubble.

5.5.2 Timing of Reionization

We now consider how the prospects for bubble detection diminish if reionization occurs earlier and the observations are focused on the corresponding redshifts. In particular, we examine the case that our model with an ionized fraction of x = 0.79 is observed at a h ii higher redshift. We focus on this case since this ionized fraction appears close to optimal for bubble detection. Aiming for only a rough estimate here, we consider the prospects for −1 detecting a RB = 40 h Mpc bubble. Although several different factors in the noise power spectrum of Equation 5.5 scale with redshift, the dominant scaling is with the sky temperature. The noise power scales as P T 2 , and the sky temperature follows T ν−2.6 (1 + z)2.6. Therefore, we expect N ∝ sky sky ∝ ∝ the signal-to-noise ratio of a detected bubble to fall off with increasing observation redshift roughly as 1+ z 2.6 S(z)= S(z ) fid . (5.14) fid 1+ z   This indicates the signal-to-noise ratio for a bubble detected with a signal-to-noise of S(zfid)

at redshift zfid = 6.9, if the bubble were instead observed at redshift z. A relatively large bubble with R 40 h−1Mpc has a typical signal-to-noise ratio at bubble center of S 4 B ≈ ≈ at our fiducial redshift. This value is found by incorporating foreground cleaning into the corresponding curve in Figure 5.4. According to Equation 5.14, the signal-to-noise value will be reduced to a significance of S 2 (1) at z = 9.3 (12.5). The bubble will, in fact, be ≈ more detectable than implied by this one number – the signal-to-noise ratio at bubble center – since an ionized region should have low signal-to-noise over much of its volume. From this, we conclude that bubble detection should be feasible with the MWA-500 if our fiducial ionized fraction occurs later than z . 9 or so, but that the prospects are rather limited in the

228 5.5 Variations on the Fiducial Model

case of significantly earlier reionization. A range of recent work in the literature, however, suggests that reionization is unlikely to complete so early. See, for example, Figure 9 from the recent study of Kuhlen and Faucher-Giguere (80) which combines Ly-α forest data (Fan et al. 52), measurements of the Thomson optical depth from WMAP (Komatsu et al. 79), and measurements of the Lyman-break galaxy luminosity function (Bouwens et al. 22). Hence, the prospects for bubble detection appear good for the MWA-500.

5.5.3 Effects of Foreground Cleaning

Next, we consider the impact of variations around our standard foreground cleaning model. As discussed previously ( 5.2.4), our standard assumption is that the impact of foreground § cleaning can be approximately mimicked by removing the running mean, over a bandwidth of B = 16 MHz, across each line of sight. The optimal foreground cleaning strategy avoids ‘over-fitting’ by removing the smoothest possible function over the largest possible band- width, in order to preserve the underlying signal as much as possible. It also avoids ‘under- fitting’ by ensuring that foreground residuals do not excessively contaminate the signal. Liu and Tegmark (92), for example, find that 21 cm foregrounds can be removed to one part in 105 or 106 by subtracting roughly three modes over 32 MHz of bandwidth. This should ∼ have a fairly similar impact to our fiducial cleaning model, but we would expect a bit more degradation in this case. A detailed investigation would add foreground contamination into our mock data cubes, and explore the impact of various cleaning algorithms directly. Here, we instead check how our results change for slightly more and less aggressive foreground cleaning. In particular, we remove the running mean over each of B = 8 MHz and B = 32 MHz and rerun our bubble finding algorithm (at z = 6.9, x = 0.79). This fid h ii has little impact on our results. In particular, the number of identified bubbles varies by less than 10 15% across the range of cleaning bandwidths considered. The quality of detected − bubbles decreases slightly for the more aggressive cleaning model, and improves slightly in the most optimistic case. Specifically, for B = 16 MHz, 96% (43%) of bubbles have ionized

229 5.5 Variations on the Fiducial Model

fractions exceeding xi = 0.79 (0.9); for B = 8 MHz the corresponding numbers are 90% (32%); and for B = 32 MHz the same numbers are 96% (62%). While these estimates are encouraging, a more detailed study is warranted. It may also be advantageous to estimate the power spectrum of the foregrounds, and incorporate this as an additional noise term in Equation 5.7 and Equation 5.9 for each of the Wiener filter and the matched filter, respectively.

5.5.4 128 Antenna Tile Configurations

So far our analysis has focused on the MWA-500, which is meant to represent a second generation 21 cm survey. In the near term, it is timely to consider the prospects for a 128 tile version of the MWA (the MWA-128) which is ramping up to take data in the very near future. This should be significantly less sensitive, since the number of baselines scales as the number of antenna tiles squared. In order to generate thermal noise representative of the MWA-128, we start by consider- ing a similar antenna distribution as for the MWA-500. In particular, we assume all of the antenna tiles are packed as closely as possible within a core of radius 8m and that the an- tenna distribution subsequently falls off as r−2 out to a maximum baseline of 1.5 km. After comparing the thermal noise power spectrum in this configuration with that in Beardsley et al. (6), we find that our noise power is larger by up to a factor of a few. This could possibly be due to our approximation of a smooth antenna distribution being less valid for the MWA-128, or to the fact that our analytic formula for the noise power spectrum does not incorporate a full treatment of rotation synthesis. In an effort to bracket somewhat the impact of the detailed antenna distribution, we further consider a configuration where all antenna tiles are packed as closely in a dense core of radius 25m. This resembles the ∼ ‘super-core’ configuration considered in Lidz et al. (91) for the power spectrum. The results of applying the optimal matched filter for a single template radius of 35h−1Mpc are shown in Figure 5.11 for the r−2 tile distribution. This shows that the

230 5.6 Favorable Antenna Configurations for Bubble Detection

sensitivity is much lower than for the MWA-500, as expected. It is much more difficult to distinguish the filtered noisy signal (bottom-left panel) from the filtered pure noise (top-left) panel here than in Figure 5.5. Most of the significant, dark blue regions in the filtered noisy signal correspond simply to low noise regions. However, applying the detection algorithm we do nonetheless detect 7 bubbles across a volume equivalent to a 6 MHz chunk of the MWA-128 survey. The success is generally lower than in the case of the MWA-500: here 75% of detected bubbles exceed the average ionized fraction of the box, while 42% exceed ∼ xi = 0.9. In the supercore configuration, we find slightly higher significance levels (up to 6.9 σ) but the identified regions generally correspond to several large clustered ionized − regions, rather than a single ionized bubble. Altogether, the algorithm identifies 10 ionized regions across the MWA survey in the supercore configuration, but the identified regions have a lower overall quality than in the r−2 configuration. Our conclusion is that bubble detection is only marginally possible with the MWA-128. While the results are unlikely to be very compelling, it is worth applying the matched filter to the first generation surveys as an initial test. Even a few weakly identified candidate bubbles would provide compelling targets for follow-up observations. Another possibility is to focus on targeted searches around known bright sources for the MWA-128 (e.g., Wyithe and Loeb 185, Friedrich et al. 57).

5.6 Favorable Antenna Configurations for Bubble Detection

The possibility of imaging or identifying ionized regions from second generation redshifted 21 cm surveys invites the question: how do we optimize future surveys for this goal? It seems unlikely that the optimal configuration for bubble detection is identical to that for measuring the power spectrum, although power spectrum measurements have mostly driven survey design considerations thus far. In the case of the power spectrum, one aims to minimize the error bar on power spectrum estimates in particular bins in wavenumber. The power spectrum error bar for each k-mode contains a thermal noise term and a sample variance

231 5.6 Favorable Antenna Configurations for Bubble Detection

(sometimes called ‘cosmic variance’) term. Because of the sample variance contribution, the gain from reducing the thermal noise for a given k-mode is limited: once the thermal noise is reduced sufficiently far below the sample variance, it is advantageous to instead measure a different k-mode on the sky within the k bin of interest. As a result, grouping | | individual antennas into only small tiles to achieve a wide survey, generally reduces the statistical error bars on power spectrum measurements compared to antenna configurations with larger tiles that probe narrower fields of view. For imaging and bubble detection, one aims for the best possible signal to noise on particular regions of the sky. In other words, for good imaging one wants to reduce the thermal noise to well below the sample variance level. Grouping individual antennas into larger tiles, in order to devote more collecting area to a narrower field of view, may be better for this purpose. In order to get some sense for these trade-offs, we consider here a LOFAR-style inter- ferometer with the specifications listed in McQuinn et al. (110). Although the detailed specifications for LOFAR have evolved somewhat (e.g., Zaroubi et al. 196), (as have the MWA specifications), this is nonetheless a helpful case to consider. In particular, our toy LOFAR-style interferometer has many fewer antenna tiles than the MWA-500 but a substan- tially larger collecting area per tile. Specifically, the interferometer considered has Na = 32 2 2 antenna tiles, Ae = 596m at our fiducial redshift (compared to Ae = 11.25m for the

MWA-500), dmin = 100m, and dmax = 2 km. We assume that antenna tiles are packed as closely as possible, consistent with dmin = 100m, inside a compact core and that the density −2 subsequently falls off as r , out to a maximum radius of rmax = 1000m. These parameters are meant to broadly represent an upgraded version of the existing LOFAR array, analogous to our MWA-500 survey, which is an upgrade to the ongoing MWA-128 instrument. With these parameters, the LOFAR-style interferometer has more total collecting area than the MWA-500 setup by a factor of a few. The results of applying a matched filter to a data cube with simulated LOFAR noise are shown in Figure 5.12. Here we zoom in to show a portion of our simulation box that matches

232 5.7 Comparisons to Previous Work

the smaller field of view of this LOFAR-like instrument. From the figure it is evident that the filter removes large scale structures, a result of the relatively large minimum baseline of this interferometer. In addition, the maximum signal-to-noise achieved here is smaller than with the MWA-500 (it drops from 10 to 6.9). Nonetheless, many small-scale ionized regions in the unfiltered noise-less signal are well preserved in the filtered noisy signal. This is consistent with the idealized calculation of Figure 5.4, which showed that LOFAR should have a higher signal-to-noise detection of small ionized regions, but a reduced signal-to-noise otherwise. Because of this, the LOFAR-style configuration is more successful during earlier stages of reionization when the bubbles are still relatively small. In general, we find that the LOFAR-style configuration detects slightly fewer bubbles overall but with more success for x . 0.79, while the MWA-500 has a greater level of success at later stages of the EoR. h ii This example suggests that the ideal configuration for bubble detection is likely interme- diate between the MWA-style and LOFAR-style antenna configurations. It appears helpful to have more collecting area on fewer baselines than the MWA, but a smaller minimum baseline than in the LOFAR-style instrument is necessary to detect large bubbles. This deserves further study, however: for example, we have neglected calibration requirements and systematic concerns. These considerations will also certainly drive the experimental design. As a further concrete example of how systematic concerns could impact the design of future arrays, suppose foreground cleaning requires removing more large scale modes than anticipated. In this case, it would make sense to focus efforts on smaller bubbles. This would shift the ideal configuration closer to a LOFAR-style instrument with a larger minimum baseline.

5.7 Comparisons to Previous Work

Previous work by Datta et al. (41) and Datta et al. (42) also considered the possibility of detecting ionized regions in noisy redshifted 21 cm data sets using a matched filter technique. The main difference between our study and this earlier work is that these previous authors

233 5.8 Conclusion

considered the prospects for detecting a specific spherical ionized region of varying size, i.e., they considered the detectability of a spherical bubble at the origin, or offset slightly from the origin. These authors also considered the case where the bubble of interest was embedded in a variety of different ionization environments; the bubble under consideration was not always isolated. Altogether, their study is mostly similar to a targeted search, where one has a good prior regarding the likely location of an ionized region. It also provides a feasibility estimate for a more ambitious blind search. The main advantage of a targeted search is that, if a region is known a priori to be highly ionized, one need not worry about an entirely false detection from a downward noise fluctuation. This then allows a lower significance threshold for robust bubble detection, and may therefore be the most feasible approach for the MWA-128 and other first generation surveys. Nonetheless, our work is a significant extension to the earlier work by Datta et al. (41) in that we conduct a blind search across an entire mock survey volume. A detailed comparison with their work is not straightforward given the difference between our approaches, but both studies have a similar bottom-line conclusion: ionized regions are detectable with surveys similar to the MWA-500.

5.8 Conclusion

We considered the prospects for making low-resolution images of the 21 cm sky and for direct, blind detection of ionized regions using first and second generation 21 cm surveys. We find that a 500-tile version of the MWA, the MWA-500, is potentially capable of detecting ionized regions. In our fiducial model, in which 79% of the volume of the Universe is ionized at z = 6.9, the MWA-500 can find 150 ionized regions in a B = 6 MHz chunk after fid ∼ 1, 000 hours of observing time. First generation surveys, such as the MWA-128, are ∼ substantially less sensitive. We find that the MWA-128 may, nonetheless, be able to detect a handful of ionized regions across its survey volume, with 7 expected in our fiducial model.

234 5.8 Conclusion

The MWA-128 may be more effective at identifying ionized bubbles using targeted searches towards, for example, bright quasars (e.g. Friedrich et al. 57). There are several possible future directions for this work. First, while we incorporate realistic levels of thermal noise and mimic the effect of foreground cleaning, it will be impor- tant to test the robustness of bubble detection with a more detailed model for foreground contamination, and to consider systematic effects from calibration errors and the MWA in- strumental response. These considerations can also help in determining the optimal design for future surveys aimed at bubble detection. Our first efforts considering which configura- tions of antenna tiles are favorable for bubble detection, detailed in 5.6, suggest that an § observing strategy intermediate to that of the MWA and LOFAR is favorable. It would also be interesting to consider the prospects for bubble identification across a larger range of reionization models than considered here. If the ionized regions at a given stage of reion- ization are, in fact, larger than in the models considered here, this should increase their detectability. On the other hand, if the ionized regions are smaller than in our present models, this would likely diminish detectability, at least for the MWA-500. If blind bubble identification is indeed feasible in future 21 cm surveys, we believe this will open up several interesting avenues of investigation. First, direct identification of ionized regions can help to build confidence in early redshifted 21 cm detections. Next, if the centers of ionized regions can be robustly identified, one may be able to use the brightness temperature contrast between the signal near the bubble’s center and its surroundings to directly constrain the cosmic mean neutral fraction (e.g., Petrovic and Oh 143). These authors also discuss how detected bubbles can be used to calibrate foreground cleaning (Petrovic and Oh 143). Finally, identifying ionized regions in redshifted 21 cm surveys allows one to commence follow-up observations, comparing galaxy properties inside detected bubbles with those in more typical regions. Typical regions and likely neutral regions can be identified as locations in the data cube with average and maximal signal-to-noise ratios, respectively, after applying the matched filter. Furthermore, if the edge of an ionized

235 5.8 Conclusion

region can be identified precisely enough, one might imagine targeted searches for galaxies at the edge of bubbles, close to neighboring neutral gas. Spectroscopic observations of these galaxies might then help to reveal the damping wing redward of the Ly-α line (e.g., Miralda-Escude 122). This would provide yet another means for constraining the neutral fraction.

236 5.8 Conclusion

Noiseless Signal Filtered Pure Noise 6.0 0.7 4 4 4.5 0.6 0.5 3.0 2 2 0.4 1.5 0.3 0 0 0.0 0.2 −1.5 -2 0.1 -2

Position (Degrees) Position (Degrees) Position −3.0 0.0 −4.5 -4 −0.1 -4 0 200 400 600 800 1000 −0.2 0 200 400 600 800 1000 −6.0 Position (h−1 Mpc) Position (h−1 Mpc)

Filtered Noisy Signal 6.0 Filtered Pure Signal 6.0 4 4 4.5 4.5

3.0 3.0 2 2 1.5 1.5

0 0.0 0 0.0

−1.5 −1.5 -2 -2

Position (Degrees) Position −3.0 (Degrees) Position −3.0

−4.5 −4.5 -4 -4 0 200 400 600 800 1000 −6.0 0 200 400 600 800 1000 −6.0 Position (h−1 Mpc) Position (h−1 Mpc)

Figure 5.2: Application of the Wiener filter to simulated data. The results are for our fiducial model with x = 0.79 at zfid = 6.9. Top-Left: Spatial slice of the unfiltered and noise-less h ii 21 cm brightness temperature contrast field (normalized by T0). Top-Right: Simulated signal- to-noise field after applying the Wiener filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the Wiener filter to the noisy signal. This can be compared with the uncorrupted input signal shown in the top-left panel and the noise realization in the top-right panel. Bottom-Right: Simulated signal-to-noise field after applying the Wiener filter to the noiseless signal. (The filtered noiseless signal shown here is normalized by the standard deviation of the noise to facilitate comparison with the other panels.) All panels show a square section of the MWA field of view transverse to the line of sight with sidelength L =1 h−1 Gpc . All slice thicknesses are 8 h−1 Mpc . Unless noted otherwise, the simulation ∼ slices in subsequent figures have these same dimensions.

237 5.8 Conclusion

150 Mpc)

1 100 − h 50

LOS ( LOS 0 0 200 400 600 800 1000 Position (h−1 Mpc)

−0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

150 Mpc)

1 100 − h 50

LOS ( LOS 0 0 200 400 600 800 1000 Position (h−1 Mpc)

−6.0 −4.5 −3.0 −1.5 0.0 1.5 3.0 4.5 6.0

Figure 5.3: Impact of foreground cleaning on the Wiener-filtered field. The top slice is a perpendicular, zoomed-in view of the simulated, unfiltered, noise-less brightness temperature contrast. The bottom slice is the signal-to-noise of the same region after applying the Wiener filter to the noisy signal field. The vertical axis shows the line-of-sight direction, with its extent −1 set to the distance scale for foreground removal, Lfg = 185 h Mpc. The horizontal axis shows a dimension transverse to the line of sight and extends 1 h−1 Gpc .

238 5.8 Conclusion

16

= 0.4 14 HI = 0.3 12 HI = 0.2 HI 10 = 0.4 (LOFAR)

)| HI B 8 |S(R 6

4

2

0 0 10 20 30 40 50 R (h−1Mpc) B

Figure 5.4: Expected signal-to-noise ratio at the center of isolated, spherical, ionized bubbles as a function of bubble radius after applying the optimal matched filter. The curves show the signal-to-noise ratio at zfid = 6.9 for the MWA-500 at various neutral fractions: xHI = 0.4 h i (blue solid), 0.3 (cyan dashed), and 0.2 (green dot-dashed). For contrast, the red dotted curve indicates the expected signal-to-noise for an interferometer with a field of view and collecting area similar to a 32-tile LOFAR-like antenna array

(at xHI =0.4). h i

239 5.8 Conclusion

Noiseless Signal Filtered Pure Noise 0.7 4 4 6 0.6 0.5 4 2 2 0.4 2 0.3 0 0 0 0.2 −2 -2 0.1 -2 Position (Degrees) Position Position (Degrees) Position 0.0 −4 −0.1 -4 -4 −6 0 200 400 600 800 1000 −0.2 0 200 400 600 800 1000 Position (h−1 Mpc) Position (h−1 Mpc)

Filtered Noisy Signal Filtered Pure Signal 4 4 6 6

4 4 2 2 2 2

0 0 0 0

−2 −2 -2 -2 Position (Degrees) Position −4 (Degrees) Position −4

-4 −6 -4 −6 0 200 400 600 800 1000 0 200 400 600 800 1000 Position (h−1 Mpc) Position (h−1 Mpc)

Figure 5.5: Application of the matched filter to simulated data and noise ( xi = 0.79 at −1 h i zfid =6.9). The template radius of the filter is 35 h Mpc , since this is a commonly detected bubble radius for our matched filter search. Top-Left: Spatial slice of the unfiltered and noise- less 21 cm brightness temperature contrast field. Top-Right: Simulated signal-to-noise field after applying the matched filter to a pure noise field. Bottom-Left: Simulated signal-to-noise field after applying the matched filter to the noisy signal. This can be compared directly to the top-left panel. Bottom-Right: Simulated signal-to-noise field after applying the matched filter to the noiseless signal. All panels are at the same spatial slice. See text for discussion on interpreting signal-to-noise values.

240 5.8 Conclusion

150 Mpc)

1 100 − h 50

LOS ( LOS 0 0 200 400 600 800 1000 Position (h−1 Mpc)

−0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

150 Mpc)

1 100 − h 50

LOS ( LOS 0 0 200 400 600 800 1000 Position (h−1 Mpc)

−6 −4 −2 0 2 4 6

Figure 5.6: Impact of foreground cleaning on the matched-filtered field. This is similar to Figure 5.3, except that the results here are for a matched filter with a template radius of −1 RT = 35 h Mpc .

241 5.8 Conclusion

Figure 5.7: An example of a detected ionized region. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the detected bubble in the matched-filtered map. Bottom-Left: Detected bubble superimposed on a zoomed-in view of the noise-less unfiltered 21 cm brightness temperature contrast map. Bottom-Right: A perpendicular zoomed-in view of the bubble depicted in the bottom-left panel. All matched- filtered maps use the template radius that minimizes the signal-to-noise at the center of the detected bubble. In the top-left case, the boxlength is L = 1 h−1 Gpc , while in the zoomed-in slices it is L 500 h−1 Mpc . ≈

242 5.8 Conclusion

Figure 5.8: An example of an ionized region that our algorithm detects as several neighboring bubbles. Top-left: Signal-to-noise field after applying the matched filter to the noisy signal. The main detected bubble is plotted on top of the corresponding region in the map. Top-Right: Zoomed-in view of the main detected bubble in the matched filtered map (solid curve) along with two other nearby detected bubbles (dashed curve). Bottom-Left: The detected bubble superimposed on the zoomed-in, noise-less, unfiltered 21 cm brightness temperature contrast map. Again, the additional nearby detected bubbles are shown (dashed curve). Bottom-Right: A perpendicular view of the bubble depicted in the bottom-left panel, with the nearby detected bubbles visible. All matched-filtered maps use the template radius that maximizes the signal to noise at the center of the main detected bubble. The box length in the top-left figure is L =1 h−1 Gpc , while in the zoomed-in panels, the box length is L = 550 h−1 Mpc .

243 5.8 Conclusion

1.1

1.0

0.9

0.8

0.7 Ionized Fraction Ionized 0.6

0.5

0.4 0 10 20 30 40 50 60 70 80 −1 RB (h Mpc)

Figure 5.9: A measure of the bubble detection success rate. The points ( ) show the volume- × averaged ionized fraction of detected bubbles versus their detected radius. For comparison, the cyan shaded region shows the 1-σ spread in the ionized fraction of randomly placed bubbles of the same radii. The bubble depicted in Fig. 5.7 is marked with a large red square, while the three bubbles shown in Fig. 5.8 are marked with large green circles.

244 5.8 Conclusion

= 0.51 = 0.68 i i 8 30

25 6 20

4 15

Frequency Frequency 10 2 5

0 0 5 12 19 26 33 40 47 54 61 68 75 5 12 19 26 33 40 47 54 61 68 75 R (h−1Mpc) R (h−1Mpc) B B = 0.79 = 0.89 i i 40 40

30 30

20 20 Frequency Frequency 10 10

0 0 5 12 19 26 33 40 47 54 61 68 75 5 12 19 26 33 40 47 54 61 68 75 R (h−1Mpc) R (h−1Mpc) B B

Figure 5.10: Size distributions of detected bubbles for varying (volume-averaged) ionization fractions. The histograms show the size distribution of (identified) ionized regions for simulation snapshots with volume-averaged ionized fractions of x =0.51 (top-left), 0.68 (top-right), 0.79 h ii (bottom-left), and 0.89 (bottom-right). These figures demonstrate how the total number and size distribution of detected bubbles varies with ionized fraction.

245 5.8 Conclusion

Noiseless Signal Filtered Pure Noise 0.7 4 4 6 0.6 0.5 4 2 2 0.4 2 0.3 0 0 0 0.2 −2 -2 0.1 -2 Position (Degrees) Position 0.0 (Degrees) Position −4 −0.1 -4 -4 −6 0 200 400 600 800 1000 −0.2 0 200 400 600 800 1000 Position (h−1 Mpc) Position (h−1 Mpc)

Filtered Noisy Signal Filtered Pure Signal 4 4 6 6

4 4 2 2 2 2

0 0 0 0

−2 −2 -2 -2 Position (Degrees) Position −4 (Degrees) Position −4

-4 −6 -4 −6 0 200 400 600 800 1000 0 200 400 600 800 1000 Position (h−1 Mpc) Position (h−1 Mpc)

Figure 5.11: Bubble detection with the MWA-128. This figure is similar to Figure 5.5, except it is for the MWA-128 configuration rather than for the MWA-500.

246 5.8 Conclusion

Noiseless Signal Filtered Pure Noise 0.7 6 1.5 0.6 1.5 1 0.5 1 4

0.5 0.4 0.5 2 0.3 0 0 0 0.2 -0.5 -0.5 0.1 −2

Position (Degrees) Position -1 (Degrees) Position -1 0.0 −4 -1.5 −0.1 -1.5 −6 0 100 200 300 400 −0.2 0 100 200 300 400 Position (h−1 Mpc) Position (h−1 Mpc)

Filtered Noisy Signal Filtered Pure Signal

1.5 6 1.5 6

1 4 1 4

0.5 2 0.5 2

0 0 0 0

-0.5 −2 -0.5 −2

Position (Degrees) Position -1 (Degrees) Position -1 −4 −4 -1.5 -1.5 −6 −6 0 100 200 300 400 0 100 200 300 400 Position (h−1 Mpc) Position (h−1 Mpc)

Figure 5.12: Bubble detection with a LOFAR-style interferometer. This figure is similar to Figure 5.5, except it is for the LOFAR configuration rather than the MWA-500. Additionally, all boxes in this figure have a side length of 426 h−1 Mpc , corresponding to the field-of-view of the LOFAR-style interferometer at z =6.9.

247 Chapter 6

Conclusion

In this thesis, we began by providing a brief overview of the field of reionization. This served several purposes. First, it gave us a general idea for what reionization is and what features viable models of reionization must share in order to reproduce observations of the high-z Universe. Furthermore, the overview recognized the importance of the enormous effort that has come before us in constraining the reionization process. However, we also emphasized that methods used in constraining the Epoch of Reioniza- tion to date invariably involve subtleties which complicate their interpretations. While the collection of observations provided by the various probes paint an important broad-stroke picture of the history of the EoR, too much credence should not be given to any single constraint. In fact, two of the most commonly-cited probes of the EoR, namely, the Ly α forest and the optical depth to Thomson scattering of CMB photons, each suffer from such compli- cations. In the case of the former, we discussed how questionable interpretations of Ly α forest measurements at z 6 have resulted in the unwarranted common knowledge that ∼ the EoR has completed by this time. In the case of the latter, revised measurements of the optical depth of CMB photons to Thomson scattering have resulted in an inferred redshift

248 of “instantaneous” reionization that has evolved substantially from zr & 15 when it was first measured to z 9 today. Recognizing this introduces the exciting realization that r ≈ reionization may be ongoing at lower redshifts than previously thought, where observations are more feasible. This provided the motivation for the remainder of the thesis. Namely, we developed ad- ditional measurement techniques that face a largely-distinct set of complications compared to the probes used to date. Along this vein, we began in Chapter 2 by developing several approaches to analyze the Ly α and Ly β forest at z 5.5 in order to constrain the EoR. ∼ We argued that, as mentioned, the common knowledge that reionization had completed by z 6 is ill-founded. As such, is is worth reconsidering what additional utility the z & 5.5 ∼ Ly α and Ly β forest might have regarding reionization. We showed that, if the Universe has a neutral fraction of x & 0.05 at z & 5.5, then damping wing absorption from neutral h HIi hydrogen and excess absorption from primordial deuterium should be observable in stacked regions of transmission in the Ly α and Ly β forest, respectively. We argued that these measurements have the added utility that they are rather direct and that the features of interest will be difficult to mimic by a highly-ionized IGM. Additionally, we showed that the presence of islands of significantly-neutral hydrogen should introduce a bimodality into the size distribution of absorbed regions in the Ly α forest at these redshifts. While this is a less direct signature of underlying neutral hydrogen, we find that the bimodality will be robust to traditional sources of complications in Ly α forest analyses, such as continuum-fitting errors. In Chapter 3, we presented the preliminary results of applying the stacking techniques mentioned above to the spectra in McGreer et al. (104). We found that an initial pass at the data did not reveal any obvious evidence for underlying islands of neutral hydrogen and that further analyses would likely yield interesting constraints on the precise neutral fraction of the IGM at z & 5.5.

249 In Chapter 4, we proposed a method for further utilizing the z & 5 Ly α forest by estimating temperature of the underlying gas. As we described in Chapter 1, the IGM should retain a thermal memory of when and how it was heated during the Epoch of Reionization. As such, measurements of the IGM temperature as close as possible to the EoR should shed light on its nature and timing. While approaches to date have typically focused on either earlier times or special regions of the IGM, the approach in Chapter 4 is applicable to typical regions of the IGM at z & 5. We demonstrated that this approach should be able to distinguish a high-z reionization scenario, ending at z 10, from a ∼ low-z reionization scenario, ending at z 6, with high statistical significance. We also ∼ demonstrated that the inhomogeneity of the EoR should be detectable in principle through its impact on the z 5 temperature field and consequently on small-scale structure in ∼ quasar spectra. Lastly, in Chapter 5, we turned to the redshifted 21-cm line from neutral hydrogen as a probe of the EoR. In principle, this line offers the ability to directly image the distribution of neutral gas throughout the Universe as a function of redshift well past the Epoch of Reionization. As such, it provides perhaps the most direct glimpse at the reionization process proposed as of yet. While such detailed imaging of the hydrogen distribution will be out of reach for the near future, we showed that it may be possible to make crude maps of the hydrogen distribution with second-generation interferometric experiments via the use of Wiener filtering. Additionally, we showed that such an instrument should be able to blindly identify the locations of hundreds of ionized regions and provide estimates ∼ of their size, thus shedding light on the progress of reionization and the ionizing sources. These approaches are very interesting in the context of experiments such as HERA, who will significantly expand on the size and capability of first-generation redshifted 21-cm experiments. The field of reionization is currently enjoying a very exciting phase. A variety of probes of the EoR have already provided tantalizing hints regarding its timing and nature. Mean-

250 while, the field is garnering attention as one of the great outstanding problems in modern astrophysics and cosmology, which motivates the acquisition of more and more data. Future high-resolution quasar observations will permit us to test for temperature inhomogeneities due to a patchy reionization along with evidence of primordial deuterium absorption as- sociated with a significantly-neutral IGM. Additionally, interferometric observations are currently underway aiming for a statistical detection of the EoR while second-generation experiments are planning to provide direct observations of the process in the near future. Such observations will not only elucidate the Epoch of Reionization, but will continue to push the observational frontiers for our Universe farther and farther back.

251 fLyα Fraction of detected galaxies which ex- hibit a strong Ly α line., page 75

Γ Decay rate for electronic transitions., page 32

γ Slope of the temperature-density rela- Glossary tion, T (δ) ≈ (1 + δ)γ−1., page 16

ΓHI Photoionization rate for hydrogen atoms. This depends on the number, location, and properties of the ionizing sources., page 15

α Fine structure constant, α ≈ 1/137, re- ~ Reduced Planck’s constant, ~ = h/2π, lated to the strength of electromagnetic the quantum of angular momentum., interactions., page 66 page 66

Ae The effective collecting area per tile in H(z) Hubble parameter., page 14 an interferometer., page 47 k Fourier wavenumber, indicative of a c Speed of light., page 14 spatial frequency., page 45

kB Boltzmann constant, page 31 δTb Brightness temperature contrast be- tween the 21-cm signal and the CMB., λα Wavelength of the Ly α transition, page 60 page 14 δ The local baryonic overdensity in units me Mass of the electron, page 14 of the cosmic mean, page 10 mp Proton mass, page 31 e Charge of the electron, page 14 ν Frequency of electromagnetic radia- hF i Mean fractional transmission, usually tion., page 24 over a region in a quasar or GRB af- terglow spectrum., page 9 n(k⊥) Number density of baselines in an in- terferometer observing k modes with fα Quantum mechanical oscillator transverse wavenumber k⊥. This de- strength for the Ly α transition, page 14 scribes the antennae configuration., page 48 fesc Fraction of ionizing photons that es-

cape their host and are injected into the nHII Local number density of ionized hydro- IGM., page 76 gen atoms., page 15

252 GLOSSARY

NHI Column number density of neutral hy- hxHIi Global volume-averaged fraction of hy- drogen atoms., page 254 drogen atoms in the neutral phase., page 26 nHI Local number density of neutral hydro- gen atoms., page 14 xα UV scattering coupling coefficient in the context of the 21-cm signal and spin Ωb Baryon density in units of the critical temperature., page 61 density, page 15 xHI Local volume-averaged neutral fraction φ(ν) Denotes the line profile for an elec- of hydrogen atoms., page 14 tronic transition, typically normalized to R dν φ(ν) = 1., page 25 XH The fraction of baryonic mass in the form of hydrogen, page 15 ρ¯ Cosmic mean baryon density., page 10 xc Collisional coupling coefficient in the ρc Critical energy density for a flat Uni- context of the 21-cm signal and spin verse, page 15 temperature., page 61

R(T ) Temperature-dependent recombination YHe The fraction of baryonic mass in the rate for ionized hydrogen., page 16 form of helium, page 15

⊙ relating to the sun (Sol), page 252 z Redshift., page 26

σ Cross section for an interaction., page 25 Bremsstrahlung Radiation Emission from

TS Spin temperature, related to the charged particles undergoing acceler- fraction of hydrogen atoms in the ation due to interactions with other triplet/singlet hyperfine state., page 49 charged particles. Literally translates to “braking radiation”., page 46 τeff The effective Gunn-Peterson optical depth, which is the negative log of the Brightness Temperature (Tb(ν)) An object mean transmission over a given redshift with a given specific intensity Iν has a corresponding brightness tempera- bin. Measurements of τeff are often con- verted into constraints on the photoion- ture equal to the requisite tempera- ization rate, page 14 ture of a blackbody for its specific intensity to equal that of the object,

τe Optical depth of CMB photons to Iν = Bν (Tb(ν))., page 38 Thomson scattering off of free elec- CMB Cosmic Microwave Background. This trons., page 66 is the earliest available snapshot of the

TK Kinetic gas temperature., page 61 Universe and is composed of light which

253 GLOSSARY

has travelled from the surface of last extremely energetic, bursts of gamma scattering to today, largely unimpeded., rays. The initial burst can last from page 2 milliseconds to several hours. They are the most energetic events known to oc- Compton Cooling Also referred to as inverse- cur in the Universe., page 8 Compton scattering, Compton cooling refers to gas that scatters off of pho- HeI Neutral helium, page 14 tons and imparts some of their kinetic HeII Singly-ionized Helium, page 14 energy on the photon. This can be an HeIII Fully-ionized helium, page 14 efficient cooling mechanism for the IGM after reionization provided it occurs at HI Neutral hydrogen, page 14 sufficiently high redshift., page 30 HII Ionized hydrogen, page 14

DLA Damped Ly α Absorber. These are Hjerting Function Also known as the Voigt dense, isolated clouds of gas with ex- Function, this function is commonly tremely high column densities of neu- used in describing line profiles which in- 20 −2 tral hydrogen, NHI & 2 × 10 cm corporate Doppler broadening and the sufficient to exhibit damping-wing ab- natural line width. This function is sorption. These are thought to source very relevant when studying the hydro- galaxy formation and are not part of the gen damping wing., page 32 diffuse IGM which we want to study for reionization purposes., page 26 IGM Intergalactic medium. This refers to the gas between galaxies, which constitutes Fourier Transform A Fourier transform essen- most of the baryonic matter in the Uni- tially describes a function as a weighted verse., page 2 sum of infinitely many sine waves kSZ Effect Kinetic Sunyaev Zel’dovich effect. with different frequencies. The Fourier This describes secondary anisotropy in transform describes the weights of the the CMB produced by the bulk veloc- different sine curves and therefore gives ities of free electrons during and af- information about the relevant dis- ter reionization which impart a Doppler tances scales in your function., page 45 shift on CMB photons., page 70 Free-Free Emission Emission from a charged Ly α Forest This describes the pattern of absorp- particle that is free both before and af- tion lines seen blueward of the rest- ter the interaction. Examples include frame Ly α line, typically in quasar electrons in an ionized hydrogen cloud spectra. These absorption lines can be interacting with protons without being due to significantly neutral gas in the captured., page 46 diffuse IGM or due to dense ionized GRB Gamma-Ray Bursts are short-lived, yet gas., page 7

254 GLOSSARY

LAE Ly α Emitter. These are galaxies which disk of a super-massive . The emit a significant fraction of their en- emission from a quasar is beamed in a ergy in the Ly α line. This line is pro- direction perpendicular to the accretion duced when hydrogen atoms within the disk., page 8 galaxy recombine after being ionized by Redshift A quantity commonly used to refer to the galaxy. Roughly 2/3 of recombina- cosmic periods of time or distances. tions result in a Ly α photon., page 72 The redshift of an object or location Lyman Series The series of transitions in an atom in space is defined as the fractional in- where an electron is transitioning to or crease in wavelength that a photon un- from the ground state., page 8 dergoes due to the expansion of the Uni- Lorentzian Distribution Probability distribu- verse while travelling from the object or tion for the ratio of two standard- location to us., page 8

normal-distributed variables. This dis- Specific Intensity (Iν ) The specific intensity of tribution also describes the intrinsic line light leaving a cloud of gas is the energy profile for absorption lines., page 24 carried by the light per unit frequency, Luminosity Function This function describes area, time, and solid angle., page 38 the luminosity distribution of sources, Synchrotron Radiation Radiation emitted by usually stars or galaxies., page 76 charged particles undergoing radial ac- Magnetic Dipole Moment The magnetic dipole celeration, such as in synchrotron parti- moment of an object is related to the cle accelerators. This constitutes a sig- torque it would experience when placed nificant source of noise for 21-cm obser- in an external magnetic field. Magnetic vations., page 46 moments are often relevant for bar mag- Voigt Function Also known as the Hjerting Func- nets or loops of current. In the con- tion, this function is commonly used in text of the hydrogen atom, the spin of describing line profiles which incorpo- the proton and electric render them as rate Doppler broadening and the nat- a sort of “loop of current” which gives ural line width. This function is very them their own magnetic dipole mo- relevant when studying the hydrogen ment., page 37 damping wing., page 32

Optical Depth Optical depth, denoted by τ, is a Wouthuysen-Field Effect Hydrogen atoms ab- quantity that describes that likelihood sorbing Ly α photons emitted from for a photon to be absorbed, usually early stars, galaxies, etc., and subse- by a gas. The fraction of photons that quently re-emitting them occasionally will pass through the gas unabsorbed is transition from singlet to triplet hyper- −τ e ., page 9 fine state., page 62 Quasar Quasars are extremely bright sources of radiation associated with the accretion

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